Factor Theorem

The Factor Theorem

For any polynomial function f(x), x − k is a factor of the polynomial if and only if f(k) = 0.

Examples

Deciding Whether x - k is a factor:

Determine whether x − 1 is a factor of each polynomial.

  1. LaTeX: f\left(x\right)=2x^4+3x^2-5x+7f(x)=2x4+3x2−5x+7
  2. LaTeX: f\left(x\right)=3x^5-2x^4+x^3-8x^2+5x+1f(x)=3x5−2x4+x3−8x2+5x+1

Solutions

  1. By the factor theorem, x − 1 will be a factor if f(1) = 0. Use synthetic division and the remainder theorem to decide.Synthetic Division to find f of 1 is 7.PNGThe remainder is 7 and not 0, so x-1 is not a factor of f(x).

  2. Again, by the factor theorem, x − 1 will be a factor if f(1) = 0. Use synthetic division and the remainder theorem to decide.
    Synthetic Division to show f of 1 is 0.PNG

Because the remainder is 0, x-1 is a factor. Additionally, we can determine from the coefficients in the bottom row that the other factor is LaTeX: 3x^4+x^{^3}+2x^2-6x-13x4+x3+2x2−6x−1

Example

Factoring a Polynomial Given a Zero

Factor LaTeX: f\left(x\right)=6x^2+19x+2x-3\:f(x)=6x2+19x+2x−3into linear factors if -3 is a zero of f.

Solution

Since -3 is a zero of f, LaTeX: x-\left(-3\right)=x+3\:x−(−3)=x+3is a factor.

Synthetic Division Dividing f by x + 3.PNG

The quotient is LaTeX: 6x^2+x-3,\:6x2+x−3,which is the factor that accompanies LaTeX: x+3.x+3.We need to factor this remaining factor to get all linear factors:

LaTeX: f\left(x\right)=6x^3+19x^2+2x-3=\left(x+3\right)\left(6x^2+x-1\right)=\left(x+3\right)\left(2x+1\right)\left(3x-1\right)f(x)=6x3+19x2+2x−3=(x+3)(6x2+x−1)=(x+3)(2x+1)(3x−1)

These factors are all linear.