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Knowledge Basic expansion and factorization

Roots of Quadratic Equations

Exponents, Logarithms and Surds

Binomial Theorem Partial Fractions

Products and Sums of Roots of PolynomialEquations

Arithmetic and Geometric Progression (JC MathSyllabus)

Sigma Notation and the Method of Difference (JCMath Syllabus)

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Products and Sums of Roots ofPolynomial Equations

Given that the roots of the quadratic equation + + 0

are and ,

+

.

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Products and Sums of Roots ofPolynomial Equations

Given that the roots of the cubic equation + + + 0

are , and ,

+ + + +

.

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Products and Sums of Roots ofPolynomial Equations

Example Exercise

Two of the roots of the equation + 3 + + + 1 0

are 1 and , for some real values of and .

What are the other two roots?

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Products and Sums of Roots ofPolynomial Equations

Let the other two roots be and . Then + + 1 12

31 +

72 .

1 12 1

11 2.

By substitution,2 + 7 4 0

12 or 4.

By symmetry, the other two roots are1

2 and 4.

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Practice Problems

(SMO 1995 Part A Question 1)

Let ,, be three real values such that 2 + 2 1 + 2 + 4 0.

Find the value of + + .(SMO 1995 Part A Question 2)

What are the real solutions of

+ 1 + 2 + 3 + 4 8?

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SMO 1995 Part A Question 1Solution

Since all three summands are nonnegative, theymust all be 0. Thus

1, 1

2

, 2,and therefore

+ + 12 .

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SMO 1995 Part A Question 2Solution

Let + . Then the equation becomes 5

2 11916 0.

Solving the quadratic equation in

andremembering that 0, we get the only solution 174 .

Hence, . Therefore

5 1 72 .

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More Practice Problems

(SMO 1996 Part A Question 3)

Let > 1 be an integer. Find all integers suchthat

+ 1

+ 1

2.

(SMO 2002 Part A Question 1)

Let be a positive integer. Evaluate, in terms of ,4 21 4

22 4

23 4

2 .

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SMO 1996 Part A Question 3

Observe that

+ 1 1 1 1

Let y + 1. Then 1.Thus,

+ 1 2

2 + 1 0 2 4

42

1

Therefore 2, 2.

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SMO 2002 Part A Question 1

The expression can be written as

4 2

=

2 6 10 4 2! 2

2 1 !

4

1 ! ! 2 !!! 2

.

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Even More Practice Problems

(SMO 2001 Part A Question 7)

Let be the real number satisfying the equation4 7 12 4 + 4 16 4 log

25

4.

Find the value of + .

(SMO 1997 Part A Question 10)Let , , , , , ( , ) be all the distinct pairs ofintegers which satisfy the equation 5 77 3.Find + + + .

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SMO 2001 Part A Question 7

Let 4, i.e., + 4.

We have

4 12 + 9 + 4 16 + 16 2 3 + 2 4 log

94 .

For RHS to be defined, y

>

.

For RHS to be non-negative, , i.e.,

< 2.

So < < 2.

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Hence 2 3 + 2 4 2 3 2 4 1.

log 94 1 + 14 7.

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SMO 1997 Part A Question 10

5 and ( 77) differ by 72. Since , areintegers and 5 77 3, it follows that( 5) and ( 77) must both be integer powersof 3, or both be the negative of integer powers of 3.

The only two integer powers of 3 that differs by 72are 9 and 81. We consider either 5 81 or

5 9, i.e.,

86or

4. Hence,

+ 86 + 4 82.

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Preview for Next Week Algebra II

Recurrence Relations

Practice Problems

Inequalities

Basic Inequalities

AM GM inequality and generalizations

Cauchy-Schwarz inequality

Practice Problems