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Remainder and Factor Theorems

Contents

  1. 1 Notes
    1. 1.1 1. Remainder Theorem
    2. 1.2 2. Factor Theorem
    3. 1.3 3. Solutions of Equations
  2. 2 Example Questions
  3. 3 Questions
  4. 4 Answers

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
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