5.5 Apply the Remainder and Factor Theorems

Polynomial long division

One method used to divide polynomials similar to the way you divide numbers

Synthetic division

A method used to divide any polynomial by a divisor of the form x - k

Example 1

Use polynomial long division

Divide 4x⁴ + 5x² - 9x + 18 by x² + 2x + 4.

Write polynomial division in the same format you use when dividing numbers. Include a “0” as the coefficient of x³. At each stage, divide the term with the highest power in what is left of the dividend by the first term of the divisor. This gives the next term of the quotient.

4x⁴ -8x³ 5x²

_x²_ _x²_ _x²_

x² + 2x + 4

_-8x³ -11x² -9x_

5x2 + 23x + 18_

_13x - 2_

You can check the result of a division problem by multiplying the quotient by the divisor and adding the remainder. The result should be the dividend.

Write the result:

= 4x² - 8x + 5 +

REMAINDER THEOREM

If a polynomial f(x) is divided by x - k, then the remainder is r = _f(k)_.

Example 2

Use synthetic division

Divide f(x) = x³ + 4x² - 5x + 3 by x + 2.

Solution

-2

_1_ 2_ -_9_ 21_

FACTOR THEOREM

A polynomial f(x) has a factor x - k if and only if f(k) =_0_.

Example 3

Factor a polynomial

Factor 2x³ - 11x² + 3x + 36 completely given that x - 3 is a factor.

Solution

Because x - 3 is a factor of f(x), you know that f(3) = _0_. Use synthetic division to find the other factors.

_3_

_2_ _-5_ _-12_ _0_

Use the result to write f(x) as a product of two factors and then factor completely.

f(x) = 2x³ - ll x + 3x + 36

=(_x - 3_) (_2x² - 5x - 12_)

=(_x - 3_) (_2x+ 3) (_x - 4_)

Example 4

Find zeros of a polynomial function

One zero of f(x) = x³ + 4x² - 15x - 18 is x = -1. Find the other zeros.

Solution

Because f(-1) = 0, _x + 1_ is a factor of f. Use synthetic division to find the other factors.

_-1_

_1_ _3_ _-18_ _0_

Use the result to write f as a product of two factors and then factor completely.

f(x) = x³ + 4x² - 15x - 18

= (_x + 1_)(_x² + 3x - 18_)

= (_x+ 1_)(_x +6_)(_x - 3_)

The zeros are _-1, -6, and 3_ .