Worksheet 5.4: Factor Theorem
Worksheet 5.4: Factor Theorem
Worksheet 5.4: Factor Theorem
Key Points
Factor theorem
Consider a polynomial f(x).
(a) If f(a) = 0, then x – a is a factor of f(x).
Conversely, if x – a is a factor of f(x), then f(a) = 0.
In each of the following, find the values of a and b. (4 – 5)
n
(b) If f = 0, then mx – n is a factor of f(x).
m 4. x 2 is a common factor of x 3 ax 2 4 x 2b and ax 3 x 2 bx 8 .
n
Conversely, if mx – n is a factor of f(x), then f = 0.
m
Note: Factor theorem is only applicable for factorizing polynomials with linear factor(s).
Use the factor theorem to determine whether each of the following is a factor of f(x).
1. f ( x) x 2 5 x 4
(a) x – 3 (b) x+2
5. 2 x 3 ax 2 bx 15 is divisible by x + 3 and x – 5.
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NSS Mathematics in Action (2nd Edition) 4A Section Worksheets 5 More about Polynomials NSS Mathematics in Action (2nd Edition) 4A Section Worksheets 5 More about Polynomials
Factorize each of the following polynomials. (6 – 11) (Use program‼) (SPYC 2020-21 S6 Mock Exam)
6. x 3 4 x 2 7 x 10 7. 3 x 3 8 x 2 41x 30 It is given that f(x) is a cubic polynomial. When f(x) is divided by x – 1 and x + 2, the remainders
are –4 and –28 respectively. When f(x) is divided by (x – 1)(x + 2), the remainder is ax + b, where
a and b are constants.
(a) Find a and b. (3 marks)
(b) It is given that x – 2 is a factor of f(x). When f(x) is divided by x, the remainder is 2.
Someone claims that all the roots of f(x) = 0 are rational numbers. Do you agree? Explain
8. x 3 3 x 2 13x 15 (!) 9. 2 x 3 24 x 2 90 x 100 (!)
your answer. (5 marks)
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