1.2 algebraic expressions
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Transcript of 1.2 algebraic expressions
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Expressions
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We order pizzas from Pizza Grande. Expressions
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We order pizzas from Pizza Grande. Each pizza is $8 and there is a $10 delivery charge.
Expressions
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We order pizzas from Pizza Grande. Each pizza is $8 and there is a $10 delivery charge. Hence if we ordered 5 pizzas delivered, the total cost would be 8(5) + 10 = $50,
Expressions
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We order pizzas from Pizza Grande. Each pizza is $8 and there is a $10 delivery charge. Hence if we ordered 5 pizzas delivered, the total cost would be 8(5) + 10 = $50, excluding the tip.
Expressions
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We order pizzas from Pizza Grande. Each pizza is $8 and there is a $10 delivery charge. Hence if we ordered 5 pizzas delivered, the total cost would be 8(5) + 10 = $50, excluding the tip.If we want x pizzas delivered, then the total cost is given by the formula “8x + 10”.
Expressions
![Page 7: 1.2 algebraic expressions](https://reader036.fdocuments.nl/reader036/viewer/2022081414/589ee4be1a28abe97f8b45a9/html5/thumbnails/7.jpg)
We order pizzas from Pizza Grande. Each pizza is $8 and there is a $10 delivery charge. Hence if we ordered 5 pizzas delivered, the total cost would be 8(5) + 10 = $50, excluding the tip.If we want x pizzas delivered, then the total cost is given by the formula “8x + 10”.Such a formula is called an expression.
Expressions
![Page 8: 1.2 algebraic expressions](https://reader036.fdocuments.nl/reader036/viewer/2022081414/589ee4be1a28abe97f8b45a9/html5/thumbnails/8.jpg)
We order pizzas from Pizza Grande. Each pizza is $8 and there is a $10 delivery charge. Hence if we ordered 5 pizzas delivered, the total cost would be 8(5) + 10 = $50, excluding the tip.If we want x pizzas delivered, then the total cost is given by the formula “8x + 10”.Such a formula is called an expression.
Expressions
If we ordered x = 100 pizzas, the cost would be 8(100)+10 = $810.
![Page 9: 1.2 algebraic expressions](https://reader036.fdocuments.nl/reader036/viewer/2022081414/589ee4be1a28abe97f8b45a9/html5/thumbnails/9.jpg)
We order pizzas from Pizza Grande. Each pizza is $8 and there is a $10 delivery charge. Hence if we ordered 5 pizzas delivered, the total cost would be 8(5) + 10 = $50, excluding the tip.If we want x pizzas delivered, then the total cost is given by the formula “8x + 10”.Such a formula is called an expression.
Expressions
If we ordered x = 100 pizzas, the cost would be 8(100)+10 = $810. The value x = 100 is called the input and the projected cost $810 is called the output.
![Page 10: 1.2 algebraic expressions](https://reader036.fdocuments.nl/reader036/viewer/2022081414/589ee4be1a28abe97f8b45a9/html5/thumbnails/10.jpg)
We order pizzas from Pizza Grande. Each pizza is $8 and there is a $10 delivery charge. Hence if we ordered 5 pizzas delivered, the total cost would be 8(5) + 10 = $50, excluding the tip.If we want x pizzas delivered, then the total cost is given by the formula “8x + 10”.Such a formula is called an expression.
Expressions
Definition: Mathematical expressions are calculation procedures which are written with numbers, variables, operation symbols +, –, *, / and ( )’s.
If we ordered x = 100 pizzas, the cost would be 8(100)+10 = $810. The value x = 100 is called the input and the projected cost $810 is called the output.
![Page 11: 1.2 algebraic expressions](https://reader036.fdocuments.nl/reader036/viewer/2022081414/589ee4be1a28abe97f8b45a9/html5/thumbnails/11.jpg)
We order pizzas from Pizza Grande. Each pizza is $8 and there is a $10 delivery charge. Hence if we ordered 5 pizzas delivered, the total cost would be 8(5) + 10 = $50, excluding the tip.If we want x pizzas delivered, then the total cost is given by the formula “8x + 10”.Such a formula is called an expression.
Expressions
Definition: Mathematical expressions are calculation procedures which are written with numbers, variables, operation symbols +, –, *, / and ( )’s. Expressions calculate future results.
If we ordered x = 100 pizzas, the cost would be 8(100)+10 = $810. The value x = 100 is called the input and the projected cost $810 is called the output.
![Page 12: 1.2 algebraic expressions](https://reader036.fdocuments.nl/reader036/viewer/2022081414/589ee4be1a28abe97f8b45a9/html5/thumbnails/12.jpg)
Algebraic Expressions
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An algebraic expression is a formula constructed with variables and numbers using addition, subtraction, multiplication, division, and taking roots.
Algebraic Expressions
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An algebraic expression is a formula constructed with variables and numbers using addition, subtraction, multiplication, division, and taking roots.
Algebraic Expressions
Examples of algebraic expressions are
3x2 – 2x + 4,
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An algebraic expression is a formula constructed with variables and numbers using addition, subtraction, multiplication, division, and taking roots.
Algebraic Expressions
Examples of algebraic expressions are
3x2 – 2x + 4,x2 + 3
3 x3 – 2x – 4 ,
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An algebraic expression is a formula constructed with variables and numbers using addition, subtraction, multiplication, division, and taking roots.
Algebraic Expressions
Examples of algebraic expressions are
3x2 – 2x + 4,x2 + 3
3 x3 – 2x – 4 ,
(x1/2 + y)1/3
(4y2 – (x + 4)1/2)1/4
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An algebraic expression is a formula constructed with variables and numbers using addition, subtraction, multiplication, division, and taking roots.
Algebraic Expressions
Examples of algebraic expressions are
3x2 – 2x + 4,x2 + 3
3 x3 – 2x – 4 ,
(x1/2 + y)1/3
(4y2 – (x + 4)1/2)1/4
Examples of non-algebraic expressions aresin(x), 2x, log(x + 1).
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An algebraic expression is a formula constructed with variables and numbers using addition, subtraction, multiplication, division, and taking roots.
Algebraic Expressions
Examples of algebraic expressions are
3x2 – 2x + 4,x2 + 3
3 x3 – 2x – 4 ,
(x1/2 + y)1/3
(4y2 – (x + 4)1/2)1/4
Examples of non-algebraic expressions aresin(x), 2x, log(x + 1).
The algebraic expressions anxn + an-1xn-1...+ a1x + a0 where ai are numbers, are called polynomials (in x).
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An algebraic expression is a formula constructed with variables and numbers using addition, subtraction, multiplication, division, and taking roots.
Algebraic Expressions
Examples of algebraic expressions are
3x2 – 2x + 4,x2 + 3
3 x3 – 2x – 4 ,
(x1/2 + y)1/3
(4y2 – (x + 4)1/2)1/4
Examples of non-algebraic expressions aresin(x), 2x, log(x + 1).
The algebraic expressions anxn + an-1xn-1...+ a1x + a0 where ai are numbers, are called polynomials (in x).
The algebraic expressions where P and Q are polynomials, are called rational expressions.
PQ
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Polynomial ExpressionsFollowing are examples of operations with polynomials and rational expressions.
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Polynomial ExpressionsFollowing are examples of operations with polynomials and rational expressions.
Example A. Expand and simplify.(2x – 5)(x +3) – (3x – 4)(x + 5)
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Polynomial ExpressionsFollowing are examples of operations with polynomials and rational expressions.
Example A. Expand and simplify.(2x – 5)(x +3) – [(3x – 4)(x + 5)] Insert [ ]
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Polynomial ExpressionsFollowing are examples of operations with polynomials and rational expressions.
Example A. Expand and simplify.(2x – 5)(x +3) – [(3x – 4)(x + 5)]= 2x2 + x – 15 – [3x2 + 11x – 20]
Insert [ ]
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Polynomial ExpressionsFollowing are examples of operations with polynomials and rational expressions.
Example A. Expand and simplify.(2x – 5)(x +3) – [(3x – 4)(x + 5)]= 2x2 + x – 15 – [3x2 + 11x – 20]= 2x2 + x – 15 – 3x2 – 11x + 20
Insert [ ]
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Polynomial ExpressionsFollowing are examples of operations with polynomials and rational expressions.
Example A. Expand and simplify.(2x – 5)(x +3) – [(3x – 4)(x + 5)]= 2x2 + x – 15 – [3x2 + 11x – 20]= 2x2 + x – 15 – 3x2 – 11x + 20= –x2 – 10x + 5
Insert [ ]
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Polynomial ExpressionsFollowing are examples of operations with polynomials and rational expressions.
Example A. Expand and simplify.(2x – 5)(x +3) – [(3x – 4)(x + 5)]= 2x2 + x – 15 – [3x2 + 11x – 20]= 2x2 + x – 15 – 3x2 – 11x + 20= –x2 – 10x + 5
Insert [ ]
To factor an expression means to write it as a product in a nontrivial way.
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Polynomial ExpressionsFollowing are examples of operations with polynomials and rational expressions.
Example A. Expand and simplify.(2x – 5)(x +3) – [(3x – 4)(x + 5)]= 2x2 + x – 15 – [3x2 + 11x – 20]= 2x2 + x – 15 – 3x2 – 11x + 20= –x2 – 10x + 5
Insert [ ]
A3 B3 = (A B)(A2 AB + B2)
Important Factoring Formulas:
To factor an expression means to write it as a product in a nontrivial way.
A2 – B2 = (A + B)(A – B)+– +– +–
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Example B. Factor 64x3 + 125Polynomial Expressions
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Example B. Factor 64x3 + 12564x3 + 125 = (4x)3 + (5)3
Polynomial Expressions
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Example B. Factor 64x3 + 12564x3 + 125 = (4x)3 + (5)3
= (4x + 5)((4x)2 – (4x)(5) +(5)2)
Polynomial Expressions
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Example B. Factor 64x3 + 12564x3 + 125 = (4x)3 + (5)3
= (4x + 5)((4x)2 – (4x)(5) +(5)2) = (4x + 5)(16x2 – 20x + 25)
Polynomial Expressions
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Example B. Factor 64x3 + 12564x3 + 125 = (4x)3 + (5)3
= (4x + 5)((4x)2 – (4x)(5) +(5)2) = (4x + 5)(16x2 – 20x + 25)
Polynomial Expressions
We factor polynomials for the following purposes.
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Example B. Factor 64x3 + 12564x3 + 125 = (4x)3 + (5)3
= (4x + 5)((4x)2 – (4x)(5) +(5)2) = (4x + 5)(16x2 – 20x + 25)
Polynomial Expressions
We factor polynomials for the following purposes.
I. It’s easier to calculate an output or to check the sign of an output using the factored form.
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Example B. Factor 64x3 + 12564x3 + 125 = (4x)3 + (5)3
= (4x + 5)((4x)2 – (4x)(5) +(5)2) = (4x + 5)(16x2 – 20x + 25)
Polynomial Expressions
We factor polynomials for the following purposes.
I. It’s easier to calculate an output or to check the sign of an output using the factored form.II. To simplify or perform algebraic operations with rational expressions.
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Example B. Factor 64x3 + 12564x3 + 125 = (4x)3 + (5)3
= (4x + 5)((4x)2 – (4x)(5) +(5)2) = (4x + 5)(16x2 – 20x + 25)
Polynomial Expressions
We factor polynomials for the following purposes.
I. It’s easier to calculate an output or to check the sign of an output using the factored form.II. To simplify or perform algebraic operations with rational expressions. III. To solve equations (See next section).
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Evaluate Polynomial ExpressionsIt's easier to evaluate factored polynomial expressions.
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Evaluate Polynomial ExpressionsIt's easier to evaluate factored polynomial expressions. It takes fewer steps then plugging in the values directly.
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Example C. Evaluate 2x3 – 5x2 + 2x for x = -2, -1, 3 by factoring it first.
Evaluate Polynomial ExpressionsIt's easier to evaluate factored polynomial expressions. It takes fewer steps then plugging in the values directly.
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Example C. Evaluate 2x3 – 5x2 + 2x for x = -2, -1, 3 by factoring it first.2x3 – 5x2 + 2x = x(2x2 – 5x + 2) = x(2x – 1)(x – 2)
Evaluate Polynomial ExpressionsIt's easier to evaluate factored polynomial expressions. It takes fewer steps then plugging in the values directly.
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Example C. Evaluate 2x3 – 5x2 + 2x for x = -2, -1, 3 by factoring it first.2x3 – 5x2 + 2x = x(2x2 – 5x + 2) = x(2x – 1)(x – 2)Plug in x = -2:-2 [2(-2) – 1] [(-2) – 2]
Evaluate Polynomial ExpressionsIt's easier to evaluate factored polynomial expressions. It takes fewer steps then plugging in the values directly.
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Example C. Evaluate 2x3 – 5x2 + 2x for x = -2, -1, 3 by factoring it first.2x3 – 5x2 + 2x = x(2x2 – 5x + 2) = x(2x – 1)(x – 2)Plug in x = -2:-2 [2(-2) – 1] [(-2) – 2] = -2 [-5] [-4] = -40
Evaluate Polynomial ExpressionsIt's easier to evaluate factored polynomial expressions. It takes fewer steps then plugging in the values directly.
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Example C. Evaluate 2x3 – 5x2 + 2x for x = -2, -1, 3 by factoring it first.2x3 – 5x2 + 2x = x(2x2 – 5x + 2) = x(2x – 1)(x – 2)Plug in x = -2:-2 [2(-2) – 1] [(-2) – 2] = -2 [-5] [-4] = -40 Plug in x = -1:-1 [2(-1) – 1] [(-1) – 2]
Evaluate Polynomial ExpressionsIt's easier to evaluate factored polynomial expressions. It takes fewer steps then plugging in the values directly.
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Example C. Evaluate 2x3 – 5x2 + 2x for x = -2, -1, 3 by factoring it first.2x3 – 5x2 + 2x = x(2x2 – 5x + 2) = x(2x – 1)(x – 2)Plug in x = -2:-2 [2(-2) – 1] [(-2) – 2] = -2 [-5] [-4] = -40 Plug in x = -1:-1 [2(-1) – 1] [(-1) – 2] = -1 [-3] [-3] = -9
Evaluate Polynomial ExpressionsIt's easier to evaluate factored polynomial expressions. It takes fewer steps then plugging in the values directly.
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Example C. Evaluate 2x3 – 5x2 + 2x for x = -2, -1, 3 by factoring it first.2x3 – 5x2 + 2x = x(2x2 – 5x + 2) = x(2x – 1)(x – 2)Plug in x = -2:-2 [2(-2) – 1] [(-2) – 2] = -2 [-5] [-4] = -40 Plug in x = -1:-1 [2(-1) – 1] [(-1) – 2] = -1 [-3] [-3] = -9 Plug in x = 3:3 [2(3) – 1] [(3) – 2]
Evaluate Polynomial ExpressionsIt's easier to evaluate factored polynomial expressions. It takes fewer steps then plugging in the values directly.
![Page 45: 1.2 algebraic expressions](https://reader036.fdocuments.nl/reader036/viewer/2022081414/589ee4be1a28abe97f8b45a9/html5/thumbnails/45.jpg)
Example C. Evaluate 2x3 – 5x2 + 2x for x = -2, -1, 3 by factoring it first.2x3 – 5x2 + 2x = x(2x2 – 5x + 2) = x(2x – 1)(x – 2)Plug in x = -2:-2 [2(-2) – 1] [(-2) – 2] = -2 [-5] [-4] = -40 Plug in x = -1:-1 [2(-1) – 1] [(-1) – 2] = -1 [-3] [-3] = -9 Plug in x = 3:3 [2(3) – 1] [(3) – 2] = 3 [5] [1] = 15
Evaluate Polynomial ExpressionsIt's easier to evaluate factored polynomial expressions. It takes fewer steps then plugging in the values directly.
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Determine the Signs of the Outputs.It's easier to determine the sign of an output, when evaluating an expression, using the factored form.
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Example D. Determine whether the outcome is + or – for x2 – 2x – 3 if x = -3/2.
Determine the Signs of the Outputs.It's easier to determine the sign of an output, when evaluating an expression, using the factored form.
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Example D. Determine whether the outcome is + or – for x2 – 2x – 3 if x = -3/2.x2 – 2x – 3 = (x – 3)(x + 1).
Determine the Signs of the Outputs.It's easier to determine the sign of an output, when evaluating an expression, using the factored form.
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Example D. Determine whether the outcome is + or – for x2 – 2x – 3 if x = -3/2.x2 – 2x – 3 = (x – 3)(x + 1). Hence for x = -3/2,we get (-3/2 – 3)(-3/2 + 1)
Determine the Signs of the Outputs.It's easier to determine the sign of an output, when evaluating an expression, using the factored form.
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Example D. Determine whether the outcome is + or – for x2 – 2x – 3 if x = -3/2.x2 – 2x – 3 = (x – 3)(x + 1). Hence for x = -3/2,we get (-3/2 – 3)(-3/2 + 1) is (–)(–) = + .
Determine the Signs of the Outputs.It's easier to determine the sign of an output, when evaluating an expression, using the factored form.
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Example D. Determine whether the outcome is + or – for x2 – 2x – 3 if x = -3/2.x2 – 2x – 3 = (x – 3)(x + 1). Hence for x = -3/2,we get (-3/2 – 3)(-3/2 + 1) is (–)(–) = + .
Determine the Signs of the Outputs.
Rational ExpressionsWe say a rational expression is in the factored formif it's numerator and denominator are factored.
It's easier to determine the sign of an output, when evaluating an expression, using the factored form.
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Example D. Determine whether the outcome is + or – for x2 – 2x – 3 if x = -3/2.x2 – 2x – 3 = (x – 3)(x + 1). Hence for x = -3/2,we get (-3/2 – 3)(-3/2 + 1) is (–)(–) = + .
Determine the Signs of the Outputs.
Rational ExpressionsWe say a rational expression is in the factored formif it's numerator and denominator are factored.
Example E. Factor
x2 – 1 x2 – 3x+ 2
It's easier to determine the sign of an output, when evaluating an expression, using the factored form.
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Example D. Determine whether the outcome is + or – for x2 – 2x – 3 if x = -3/2.x2 – 2x – 3 = (x – 3)(x + 1). Hence for x = -3/2,we get (-3/2 – 3)(-3/2 + 1) is (–)(–) = + .
Determine the Signs of the Outputs.
Rational ExpressionsWe say a rational expression is in the factored formif it's numerator and denominator are factored.
Example E. Factor
x2 – 1 x2 – 3x+ 2
x2 – 1 x2 – 3x+ 2
= (x – 1)(x + 1) (x – 1)(x – 2)
is the factored form.
It's easier to determine the sign of an output, when evaluating an expression, using the factored form.
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Rational ExpressionsWe put rational expressions in the factored form in order to reduce, multiply or divide them.
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Rational ExpressionsWe put rational expressions in the factored form in order to reduce, multiply or divide them.
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Rational ExpressionsWe put rational expressions in the factored form in order to reduce, multiply or divide them. Cancellation Rule: Given a rational expression in the factored form, common factors may be cancelled,
i.e. x*yx*z = x*y
x*z = yz
1
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Rational ExpressionsWe put rational expressions in the factored form in order to reduce, multiply or divide them.
x*yx*z = x*y
x*z = yz
A rational expression that can't be cancelled any further is said to be reduced.
Cancellation Rule: Given a rational expression in the factored form, common factors may be cancelled,
i.e.
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Rational ExpressionsWe put rational expressions in the factored form in order to reduce, multiply or divide them.
Example F. Reduce x2 – 1 x2 – 3x+ 2
x*yx*z = x*y
x*z = yz
A rational expression that can't be cancelled any further is said to be reduced.
Cancellation Rule: Given a rational expression in the factored form, common factors may be cancelled,
i.e.
![Page 59: 1.2 algebraic expressions](https://reader036.fdocuments.nl/reader036/viewer/2022081414/589ee4be1a28abe97f8b45a9/html5/thumbnails/59.jpg)
Rational ExpressionsWe put rational expressions in the factored form in order to reduce, multiply or divide them.
Example F. Reduce x2 – 1 x2 – 3x+ 2
x2 – 1 x2 – 3x+ 2 = (x – 1)(x + 1)
(x – 1)(x – 2)
x*yx*z = x*y
x*z = yz
A rational expression that can't be cancelled any further is said to be reduced.
factor
Cancellation Rule: Given a rational expression in the factored form, common factors may be cancelled,
i.e.
![Page 60: 1.2 algebraic expressions](https://reader036.fdocuments.nl/reader036/viewer/2022081414/589ee4be1a28abe97f8b45a9/html5/thumbnails/60.jpg)
Rational ExpressionsWe put rational expressions in the factored form in order to reduce, multiply or divide them.
Example F. Reduce x2 – 1 x2 – 3x+ 2
x2 – 1 x2 – 3x+ 2 = (x – 1)(x + 1)
(x – 1)(x – 2)
x*yx*z = x*y
x*z = yz
A rational expression that can't be cancelled any further is said to be reduced.
= (x + 1) (x – 2)
factor
Cancellation Rule: Given a rational expression in the factored form, common factors may be cancelled,
i.e.
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Rational ExpressionsMultiplication Rule: PQ
RS* = P*R
Q*S
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Rational ExpressionsMultiplication Rule: PQ
RS* = P*R
Q*S
Division Rule: PQ
RS
÷ = P*SQ*R
Reciprocate
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Rational ExpressionsMultiplication Rule:
To carry out these operations, put the expressions in factored form and cancel as much as possible.
PQ
RS* = P*R
Q*S
Division Rule: PQ
RS
÷ = P*SQ*R
Reciprocate
![Page 64: 1.2 algebraic expressions](https://reader036.fdocuments.nl/reader036/viewer/2022081414/589ee4be1a28abe97f8b45a9/html5/thumbnails/64.jpg)
Rational ExpressionsMultiplication Rule:
To carry out these operations, put the expressions in factored form and cancel as much as possible.
PQ
RS* = P*R
Q*S
Division Rule: PQ
RS
÷ = P*SQ*R
Reciprocate
Example G. Simplify
(2x – 6) (y + 3) ÷ (y2 + 2y – 3)
(9 – x2)
![Page 65: 1.2 algebraic expressions](https://reader036.fdocuments.nl/reader036/viewer/2022081414/589ee4be1a28abe97f8b45a9/html5/thumbnails/65.jpg)
Rational ExpressionsMultiplication Rule:
To carry out these operations, put the expressions in factored form and cancel as much as possible.
PQ
RS* = P*R
Q*S
Division Rule: PQ
RS
÷ = P*SQ*R
Reciprocate
Example G. Simplify
(2x – 6) (y + 3) ÷ (y2 + 2y – 3)
(9 – x2)
(2x – 6) (y + 3) ÷ (y2 + 2y – 3)
(9 – x2) = (2x – 6) (y + 3)
(y2 + 2y – 3) (9 – x2)*
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Rational ExpressionsMultiplication Rule:
To carry out these operations, put the expressions in factored form and cancel as much as possible.
PQ
RS* = P*R
Q*S
Division Rule: PQ
RS
÷ = P*SQ*R
Reciprocate
Example G. Simplify
(2x – 6) (y + 3) ÷ (y2 + 2y – 3)
(9 – x2)
(2x – 6) (y + 3) ÷ (y2 + 2y – 3)
(9 – x2) = (2x – 6) (y + 3)
(y2 + 2y – 3) (9 – x2)*
= 2(x – 3) (y + 3)
(y + 3)(y – 1) (3 – x)(3 + x)*
![Page 67: 1.2 algebraic expressions](https://reader036.fdocuments.nl/reader036/viewer/2022081414/589ee4be1a28abe97f8b45a9/html5/thumbnails/67.jpg)
Rational ExpressionsMultiplication Rule:
To carry out these operations, put the expressions in factored form and cancel as much as possible.
PQ
RS* = P*R
Q*S
Division Rule: PQ
RS
÷ = P*SQ*R
Reciprocate
Example G. Simplify
(2x – 6) (y + 3) ÷ (y2 + 2y – 3)
(9 – x2)
(2x – 6) (y + 3) ÷ (y2 + 2y – 3)
(9 – x2) = (2x – 6) (y + 3)
(y2 + 2y – 3) (9 – x2)*
= 2(x – 3) (y + 3)
(y + 3)(y – 1) (3 – x)(3 + x)*
1
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Rational ExpressionsMultiplication Rule:
To carry out these operations, put the expressions in factored form and cancel as much as possible.
PQ
RS* = P*R
Q*S
Division Rule: PQ
RS
÷ = P*SQ*R
Reciprocate
Example G. Simplify
(2x – 6) (y + 3) ÷ (y2 + 2y – 3)
(9 – x2)
(2x – 6) (y + 3) ÷ (y2 + 2y – 3)
(9 – x2) = (2x – 6) (y + 3)
(y2 + 2y – 3) (9 – x2)*
= 2(x – 3) (y + 3)
(y + 3)(y – 1) (3 – x)(3 + x)*
–1 1
= –2(y – 1) (x + 3)
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Rational ExpressionsThe least common denominator (LCD) is needed
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Rational ExpressionsThe least common denominator (LCD) is needed I. to combine (add or subtract) rational expressions
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Rational ExpressionsThe least common denominator (LCD) is needed I. to combine (add or subtract) rational expressionsII. to simplify complex fractions
![Page 72: 1.2 algebraic expressions](https://reader036.fdocuments.nl/reader036/viewer/2022081414/589ee4be1a28abe97f8b45a9/html5/thumbnails/72.jpg)
Rational ExpressionsThe least common denominator (LCD) is needed I. to combine (add or subtract) rational expressionsII. to simplify complex fractionsIn all these applications, use the LCD as a multiplier to clear denominators.
![Page 73: 1.2 algebraic expressions](https://reader036.fdocuments.nl/reader036/viewer/2022081414/589ee4be1a28abe97f8b45a9/html5/thumbnails/73.jpg)
Rational ExpressionsThe least common denominator (LCD) is needed I. to combine (add or subtract) rational expressionsII. to simplify complex fractionsIn all these applications, use the LCD as a multiplier to clear denominators. To Combine Rational Expressions (LCD Method):
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Rational ExpressionsThe least common denominator (LCD) is needed I. to combine (add or subtract) rational expressionsII. to simplify complex fractionsIn all these applications, use the LCD as a multiplier to clear denominators. To Combine Rational Expressions (LCD Method): To combine rational expressions (F ± G), the numerator of the answer is (LCD)(F ± G) i.e.the answer is (LCD)(F ± G) / LCD.
![Page 75: 1.2 algebraic expressions](https://reader036.fdocuments.nl/reader036/viewer/2022081414/589ee4be1a28abe97f8b45a9/html5/thumbnails/75.jpg)
Rational ExpressionsThe least common denominator (LCD) is needed I. to combine (add or subtract) rational expressionsII. to simplify complex fractionsIn all these applications, use the LCD as a multiplier to clear denominators. To Combine Rational Expressions (LCD Method): To combine rational expressions (F ± G), the numerator of the answer is (LCD)(F ± G) i.e.the answer is (LCD)(F ± G) / LCD.
Example H: Combine 712
58 + –
169
![Page 76: 1.2 algebraic expressions](https://reader036.fdocuments.nl/reader036/viewer/2022081414/589ee4be1a28abe97f8b45a9/html5/thumbnails/76.jpg)
Rational ExpressionsThe least common denominator (LCD) is needed I. to combine (add or subtract) rational expressionsII. to simplify complex fractionsIn all these applications, use the LCD as a multiplier to clear denominators. To Combine Rational Expressions (LCD Method): To combine rational expressions (F ± G), the numerator of the answer is (LCD)(F ± G) i.e.the answer is (LCD)(F ± G) / LCD.
Example H: Combine 712
58 + –
169
The LCD = 48,
![Page 77: 1.2 algebraic expressions](https://reader036.fdocuments.nl/reader036/viewer/2022081414/589ee4be1a28abe97f8b45a9/html5/thumbnails/77.jpg)
Rational ExpressionsThe least common denominator (LCD) is needed I. to combine (add or subtract) rational expressionsII. to simplify complex fractionsIn all these applications, use the LCD as a multiplier to clear denominators. To Combine Rational Expressions (LCD Method): To combine rational expressions (F ± G), the numerator of the answer is (LCD)(F ± G) i.e.the answer is (LCD)(F ± G) / LCD.
Example H: Combine 712
58 + –
169
The LCD = 48, ( ) *48 7
1258
+ – 169
![Page 78: 1.2 algebraic expressions](https://reader036.fdocuments.nl/reader036/viewer/2022081414/589ee4be1a28abe97f8b45a9/html5/thumbnails/78.jpg)
Rational ExpressionsThe least common denominator (LCD) is needed I. to combine (add or subtract) rational expressionsII. to simplify complex fractionsIn all these applications, use the LCD as a multiplier to clear denominators. To Combine Rational Expressions (LCD Method): To combine rational expressions (F ± G), the numerator of the answer is (LCD)(F ± G) i.e.the answer is (LCD)(F ± G) / LCD.
Example H: Combine 712
58 + –
169
The LCD = 48, ( ) *48 67
1258
+ – 1694 3
![Page 79: 1.2 algebraic expressions](https://reader036.fdocuments.nl/reader036/viewer/2022081414/589ee4be1a28abe97f8b45a9/html5/thumbnails/79.jpg)
Rational ExpressionsThe least common denominator (LCD) is needed I. to combine (add or subtract) rational expressionsII. to simplify complex fractionsIn all these applications, use the LCD as a multiplier to clear denominators. To Combine Rational Expressions (LCD Method): To combine rational expressions (F ± G), the numerator of the answer is (LCD)(F ± G) i.e.the answer is (LCD)(F ± G) / LCD.
Example H: Combine 712
58 + –
169
The LCD = 48, ( ) *48 67
1258
+ – 1694 3
= 28 + 30 – 27 = 31
![Page 80: 1.2 algebraic expressions](https://reader036.fdocuments.nl/reader036/viewer/2022081414/589ee4be1a28abe97f8b45a9/html5/thumbnails/80.jpg)
Rational ExpressionsThe least common denominator (LCD) is needed I. to combine (add or subtract) rational expressionsII. to simplify complex fractionsIn all these applications, use the LCD as a multiplier to clear denominators. To Combine Rational Expressions (LCD Method): To combine rational expressions (F ± G), the numerator of the answer is (LCD)(F ± G) i.e.the answer is (LCD)(F ± G) / LCD.
Example H: Combine 712
58 + –
169
The LCD = 48, ( ) *48 67
1258
+ – 1694 3
= 28 + 30 – 27 = 31
Hence 712
58 + –
169 =
4831
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Rational Expressions
– (y2 + 2y – 3) (y2 + y – 2) 2y – 1 y – 3 Example I.
Combine
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Rational Expressions
– (y2 + 2y – 3) (y2 + y – 2) 2y – 1 y – 3
y2 + y – 2 = (y – 1)(y + 2)
Example I. Combine
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Rational Expressions
– (y2 + 2y – 3) (y2 + y – 2) 2y – 1 y – 3
y2 + y – 2 = (y – 1)(y + 2)
y2 + 2y – 3 = (y – 1)(y + 3)
Example I. Combine
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Rational Expressions
– (y2 + 2y – 3) (y2 + y – 2) 2y – 1 y – 3
y2 + y – 2 = (y – 1)(y + 2)
y2 + 2y – 3 = (y – 1)(y + 3)
Hence the LCD = (y – 1)(y + 2)(y + 3),
Example I. Combine
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Rational Expressions
– (y2 + 2y – 3) (y2 + y – 2) 2y – 1 y – 3
y2 + y – 2 = (y – 1)(y + 2)
y2 + 2y – 3 = (y – 1)(y + 3)
Hence the LCD = (y – 1)(y + 2)(y + 3), multiply it to the problem:
– (y2 + 2y – 3) (y – 1)(y + 2) 2y – 1 y – 3 [ ] (y – 1)(y + 2)(y + 3)
Example I. Combine
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Rational ExpressionsExample I. Combine
– (y2 + 2y – 3) (y2 + y – 2) 2y – 1 y – 3
y2 + y – 2 = (y – 1)(y + 2)
y2 + 2y – 3 = (y – 1)(y + 3)
Hence the LCD = (y – 1)(y + 2)(y + 3), multiply it to the problem:
– (y2 + 2y – 3) (y – 1)(y + 2) 2y – 1 y – 3 [ ] (y – 1)(y + 2)(y + 3)
(y + 3) (y + 2)
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Rational Expressions
– (y2 + 2y – 3) (y2 + y – 2) 2y – 1 y – 3
y2 + y – 2 = (y – 1)(y + 2)
y2 + 2y – 3 = (y – 1)(y + 3)
Hence the LCD = (y – 1)(y + 2)(y + 3), multiply it to the problem:
– (y2 + 2y – 3) (y – 1)(y + 2) 2y – 1 y – 3 [ ] (y – 1)(y + 2)(y + 3)
= (2y – 1)(y + 3) – (y – 3)(y + 2)
(y + 3) (y + 2)
Example I. Combine
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Rational Expressions
– (y2 + 2y – 3) (y2 + y – 2) 2y – 1 y – 3
y2 + y – 2 = (y – 1)(y + 2)
y2 + 2y – 3 = (y – 1)(y + 3)
Hence the LCD = (y – 1)(y + 2)(y + 3), multiply it to the problem:
– (y2 + 2y – 3) (y – 1)(y + 2) 2y – 1 y – 3 [ ] (y – 1)(y + 2)(y + 3)
= (2y – 1)(y + 3) – (y – 3)(y + 2) = y2 + 6y + 3
(y + 3) (y + 2)
Example I. Combine
![Page 89: 1.2 algebraic expressions](https://reader036.fdocuments.nl/reader036/viewer/2022081414/589ee4be1a28abe97f8b45a9/html5/thumbnails/89.jpg)
Rational Expressions
– (y2 + 2y – 3) (y2 + y – 2) 2y – 1 y – 3
y2 + y – 2 = (y – 1)(y + 2)
y2 + 2y – 3 = (y – 1)(y + 3)
Hence the LCD = (y – 1)(y + 2)(y + 3), multiply it to the problem:
– (y2 + 2y – 3) (y – 1)(y + 2) 2y – 1 y – 3 [ ] (y – 1)(y + 2)(y + 3)
= (2y – 1)(y + 3) – (y – 3)(y + 2) = y2 + 6y + 3
So – (y2 + 2y – 3) (y2 + y – 2) 2y – 1 y – 3 = y2 + 6y + 3
(y – 1)(y + 2)(y + 3)
(y + 3) (y + 2)
![Page 90: 1.2 algebraic expressions](https://reader036.fdocuments.nl/reader036/viewer/2022081414/589ee4be1a28abe97f8b45a9/html5/thumbnails/90.jpg)
Rational ExpressionsA complex fraction is a fraction made with rational expressions.
![Page 91: 1.2 algebraic expressions](https://reader036.fdocuments.nl/reader036/viewer/2022081414/589ee4be1a28abe97f8b45a9/html5/thumbnails/91.jpg)
Rational ExpressionsA complex fraction is a fraction made with rational expressions. To simplify a complex fraction, use thethe LCD to clear all denominators.
![Page 92: 1.2 algebraic expressions](https://reader036.fdocuments.nl/reader036/viewer/2022081414/589ee4be1a28abe97f8b45a9/html5/thumbnails/92.jpg)
Rational Expressions
Example J. Simplify –(x – h)
1
A complex fraction is a fraction made with rational expressions. To simplify a complex fraction, use thethe LCD to clear all denominators.
(x + h)
1
2h
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Rational Expressions
Example J. Simplify –(x – h)
1
A complex fraction is a fraction made with rational expressions. To simplify a complex fraction, use thethe LCD to clear all denominators.
(x + h)
1
2h Multiply the top and bottom by (x – h)(x + h) to reduce the expression in the numerators to polynomials.
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Rational Expressions
Example J. Simplify –(x – h)
1
A complex fraction is a fraction made with rational expressions. To simplify a complex fraction, use thethe LCD to clear all denominators.
(x + h)
1
2h Multiply the top and bottom by (x – h)(x + h) to reduce the expression in the numerators to polynomials. –(x – h)
1 (x + h)
1
2h = –(x – h)
1 (x + h)
1
2h
(x + h)(x – h)
[
] (x + h)(x –
h)
*
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Rational Expressions
Example J. Simplify –(x – h)
1
A complex fraction is a fraction made with rational expressions. To simplify a complex fraction, use thethe LCD to clear all denominators.
(x + h)
1
2h Multiply the top and bottom by (x – h)(x + h) to reduce the expression in the numerators to polynomials. –(x – h)
1 (x + h)
1
2h = –(x – h)
1 (x + h)
1
2h
(x + h)(x – h)
[
] (x + h)(x –
h)
*
= –(x + h)
(x – h) 2h (x + h)(x –
h)
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Rational Expressions
Example J. Simplify –(x – h)
1
A complex fraction is a fraction made with rational expressions. To simplify a complex fraction, use thethe LCD to clear all denominators.
(x + h)
1
2h Multiply the top and bottom by (x – h)(x + h) to reduce the expression in the numerators to polynomials. –(x – h)
1 (x + h)
1
2h = –(x – h)
1 (x + h)
1
2h
(x + h)(x – h)
[
] (x + h)(x –
h)
*
= –(x + h)
(x – h) 2h (x + h)(x –
h) = 2h
2h (x + h)(x – h)
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Rational Expressions
Example J. Simplify –(x – h)
1
A complex fraction is a fraction made with rational expressions. To simplify a complex fraction, use thethe LCD to clear all denominators.
(x + h)
1
2h Multiply the top and bottom by (x – h)(x + h) to reduce the expression in the numerators to polynomials. –(x – h)
1 (x + h)
1
2h = –(x – h)
1 (x + h)
1
2h
(x + h)(x – h)
[
] (x + h)(x –
h)
*
= –(x + h)
(x – h) 2h (x + h)(x –
h) = 2h
2h (x + h)(x – h)
= 1 (x + h)(x – h)
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To rationalize radicals in expressions we often use the formula (x – y)(x + y) = x2 – y2.
Rationalize Radicals
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To rationalize radicals in expressions we often use the formula (x – y)(x + y) = x2 – y2. (x + y) and (x – y) are called conjugates.
Rationalize Radicals
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Example K: Rationalize the numerator
To rationalize radicals in expressions we often use the formula (x – y)(x + y) = x2 – y2. (x + y) and (x – y) are called conjugates.
hx + h – x
Rationalize Radicals
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To rationalize radicals in expressions we often use the formula (x – y)(x + y) = x2 – y2. (x + y) and (x – y) are called conjugates.
Rationalize Radicals
hx + h – x = h
(x + h – x) (x + h + x) (x + h + x)
*
Example K: Rationalize the numerator hx + h – x
![Page 102: 1.2 algebraic expressions](https://reader036.fdocuments.nl/reader036/viewer/2022081414/589ee4be1a28abe97f8b45a9/html5/thumbnails/102.jpg)
To rationalize radicals in expressions we often use the formula (x – y)(x + y) = x2 – y2. (x + y) and (x – y) are called conjugates.
Rationalize Radicals
hx + h – x = h
(x + h – x) (x + h + x) (x + h + x)
*
=h
(x + h)2 – (x)2 (x + h + x)
Example K: Rationalize the numerator hx + h – x
![Page 103: 1.2 algebraic expressions](https://reader036.fdocuments.nl/reader036/viewer/2022081414/589ee4be1a28abe97f8b45a9/html5/thumbnails/103.jpg)
To rationalize radicals in expressions we often use the formula (x – y)(x + y) = x2 – y2. (x + y) and (x – y) are called conjugates.
Rationalize Radicals
hx + h – x = h
(x + h – x) (x + h + x) (x + h + x)
*
=h
(x + h)2 – (x)2 (x + h + x)
Example K: Rationalize the numerator hx + h – x
(x + h) – (x) = h
![Page 104: 1.2 algebraic expressions](https://reader036.fdocuments.nl/reader036/viewer/2022081414/589ee4be1a28abe97f8b45a9/html5/thumbnails/104.jpg)
To rationalize radicals in expressions we often use the formula (x – y)(x + y) = x2 – y2. (x + y) and (x – y) are called conjugates.
Rationalize Radicals
hx + h – x = h
(x + h – x) (x + h + x) (x + h + x)
*
=h
(x + h)2 – (x)2 (x + h + x)
=h
h(x + h + x)
Example K: Rationalize the numerator hx + h – x
(x + h) – (x) = h
![Page 105: 1.2 algebraic expressions](https://reader036.fdocuments.nl/reader036/viewer/2022081414/589ee4be1a28abe97f8b45a9/html5/thumbnails/105.jpg)
To rationalize radicals in expressions we often use the formula (x – y)(x + y) = x2 – y2. (x + y) and (x – y) are called conjugates.
Rationalize Radicals
hx + h – x = h
(x + h – x) (x + h + x) (x + h + x)
*
=h
(x + h)2 – (x)2 (x + h + x)
=h
h(x + h + x)
=1
x + h + x
Example K: Rationalize the numerator hx + h – x
(x + h) – (x) = h
![Page 106: 1.2 algebraic expressions](https://reader036.fdocuments.nl/reader036/viewer/2022081414/589ee4be1a28abe97f8b45a9/html5/thumbnails/106.jpg)
Algebraic ExpressionsExercise 1.2 AExpand1. –(2x – 5)(x +3) 2. (3x – 4)(x + 5) – (2x – 5)(x + 3)3. (3x – 4)(2x – 5) – (x + 5)(x + 3)4. (3x – 4)(x + 3) – (x + 5)(2x – 5)Evaluate if x = -3, -2, -1 by using the factored form5. 3x2 – 2x – 1 6. 2x2 – 5x + 27. 4x3 + 3x2 – x 8 . -2x4 + 3x3 + 2x2 Find the signs of the output if x = -3/2, 2/3 9. 3x2 – 2x – 1 10. 2x2 – 5x + 211. 4x3 + 3x2 – x 12 . -2x4 + 3x3 + 2x2
![Page 107: 1.2 algebraic expressions](https://reader036.fdocuments.nl/reader036/viewer/2022081414/589ee4be1a28abe97f8b45a9/html5/thumbnails/107.jpg)
Algebraic Expressions