Remainder and Factor theorem

Section 2-2: Remainder and Factor Theorems

Two theorems are important when dealing with synthetic substitution.

1) The remainder Theorem

2) The factor Theorem

The remainder theorem: When a polynomial P(x) is divided by x - a , the remainder is P(a)! Remember basic arithmetic: Dividend = Divisor x Quotient + Remainder If we apply this to synthetic division, we get the following result: Divide: x³ + 5x² + 5x - 3 by x + 2 keep in mind it must be in the form x-(-2)!

    Use synthetic division: -2|            1        5        5        -3
                                                                    -2       -6         2
                                                           1        3       -1        -1
                                                            x² + 3x - 1            -1
                                                            Quotient        Remainder
                    The correct form of the answer: x² + 3x - 1 - 1/(x + 2)

The factor theorem: For a polynomial P(x), x - a is a factor iff P(a) = 0. This says that if using synthetic division, the divisor is a factor (also the quotient) when the last number is zero! If the last number isn't zero, what you divided by is not a factor and neither is the quotient!

Sample problems! 1) Find the quotient and remainder of x³ + 8x² -5x - 84 divided by x + 5

2) Determine if x - 3 is a factor of x³ + 5x² - 17x - 21

3) Show that x + 2 is a factor of 2x³ + 3x² - 8x - 12 and then factor completely.

4) Show that x + 3 and x - 4 are factors of x⁴ - 2x³ - 13x² + 14x + 24 and then factor completely.
Answers are at the bottom of this page!

Answers 1) -5| 1 8 -5 -84 -5 -15 100 1 3 -20 16 x² + 3x - 20 + 16/(x + 5) 2) 3| 1 5 -17 -21 3 24 21 1 8 7 0 Since the remainder is 0, the divisor is a factor! 3) -2| 2 3 -8 -12 -4 2 12 2 -1 -6 0 (2x² - x - 6)(x + 2) = (2x + 3)(x - 2)(x + 2) 4) -3| 1 -2 -13 14 24 -3 15 -6 -24 1 -5 2 8 0 4| 1 -5 2 8 4 -4 -8 1 -1 -2 0 (x² - x - 2)(x + 3)(x - 4) = (x -2)(x + 1)(x + 3)(x - 4)