Using the Factor Theorem

6.3.5 Using the Factor Theorem

6.3.5 • Using the Factor Theorem

Cool Down

In all the division problems we completed that had a remainder of zero, we say that the divisor was a factor of the dividend.

Whenever a divisor, x−cx−c, divides a polynomial function, f(x)f(x), and results in a remainder of zero, we say x−cx−c is a factor of f(x)f(x).

This leads us to the Factor Theorem.

FACTOR THEOREM

For any polynomial function f(x)f(x),

  • if x−cx−c is a factor of f(x)f(x), then f(c)=0f(c)=0.
  • if f(c)=0f(c)=0, then x−cx−c is a factor of f(x)f(x).

Use the Factor Theorem to determine whether the binomial is a factor of the given polynomial function. Be prepared to show your reasoning.

1.

Is (x−5)(x−5) a factor of f(x)=x3−125f(x)=x3−125?

Compare your answer:

Yes, because f(5)=(5)3−125=125−125=0f(5)=(5)3−125=125−125=0

2.

Is (x−6)(x−6) a factor of f(x)=x3−216f(x)=x3−216?

Compare your answer:

Yes, because f(6)=(6)3−216=216−216=0f(6)=(6)3−216=216−216=0

3.

Is (x+3)(x+3) a factor of f(x)=x3+8x2+21x+18f(x)=x3+8x2+21x+18?

Compare your answer:

Yes, because f(−3)=(−3)3+8(−3)2+21(−3)+18=−27+72−63+18=0f(−3)=(−3)3+8(−3)2+21(−3)+18=−27+72−63+18=0

4.

Is (x+4)(x+4) a factor of f(x)=x3+x2−14x+8f(x)=x3+x2−14x+8?

Compare your answer:

No, because f(−4)=(−4)3+(−4)2−14(−4)+8=−64+16+56+8=16f(−4)=(−4)3+(−4)2−14(−4)+8=−64+16+56+8=16