What is the Factor Theorem? Definition and Examples Showing How to Use it


The factor theorem is a theorem in algebra that provides a method for determining if a polynomial has a specific linear factor.

Factor theorem:

Let p(x) be a polynomial function. The expression x - a is a linear factor of p(x) if and only if the value a is a zero of p(x).

In other words,

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

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

How to Use the Factor Theorem

Suppose the zero(s) of a polynomial function is(are) known. You can use the zero(s) to find the standard form of the polynomial function.

Example #1

The zeros of a polynomial function are -2 and 5. Find the standard form of the polynomial function.

Since -2 is a zero of the function, x - -2 or x + 2 is a linear factor.

Since 5 is a zero of the function, x - 5 is a linear factor.

Just multiply the linear factors to find the standard form of the polynomial function.

(x + 2)(x - 5) = x2 -5x + 2x - 10

(x + 2)(x - 5) = x2 -3x - 10

Suppose a zero of a polynomial function is known. You can use the zero that is known to find other zero(s) of the polynomial function.

Example #2

Suppose -4 is a zero of x2 + 3x - 4. Find the other zero.

If -4 is a zero, then x + 4 is a linear factor. Let x - a be the other factor.

(x + 4)(x - a) = x2 + 3x - 4

x2 + -ax + 4x - 4a = x2 + 3x - 4

x2 + (-a + 4)x - 4a = x2 + 3x - 4

Notice that -a + 4 must be equal to 3.

-a + 4 = 3

a = 1 since -1 + 4 = 3

The other zero is a  = 1

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