Remainder and Factor Theorems

Notes

1. Remainder Theorem

If a polynomial f(x) is divided by x - a, then the remainder, R = f(a).

2. Factor Theorem

If f(a) = 0, then (x - a) is a factor of f(x)

3. Solutions of Equations

To solve the equation f(x) = 0, first factorize f(x) by the Factor Theorem

Eg . solve x³ - 2x² - 5x + 6 = 0

Let f(x) = x³ - 2x² - 5x + 6

Find a factor by trial and error

eg, found that (x - 1) is a factor

Divide f(x) by (x - 1) and we get

f(x) = (x - 1)(x² - x - 6)

Factorize x² - x - 6

---> f(x) = (x - 1)(x + 2)(x - 3)

so, (x - 1)(x + 2)(x - 3) = 0

so x = 1, -2, or 3

Example Questions

Note: How to divide f(x) by (x + 1)

Note: How to factorize 4x² -12x +9

Questions

Answers

1a. -2, 3, 5

1b. 1, -2, -1/2

1c. 1/2, -1/2, 2/3

2. 2

3a. a = 3, b = 8

3b. (2x - 1)(x - 1)

4. -2

5. c = 0 , 3, -3