2.7.1 Factor and Remainder Theorem

Factor Theorem

• The factor theorem is used to find the linear factors of polynomial equations

• This topic is closely tied to finding the zeros and roots of a polynomial function/equation

  • As a rule of thumb a zero refers to the polynomial function and a root refers to a polynomial equation

Factor Theorem Formula

• For any polynomial function P(x)

  • (x - k) is a factor of P(x) if P(k) = 0

  • P(k) = 0 if (x - k) is a factor of P(x)

Using the Factor Theorem

If the linear factor has a coefficient of ‘x’, then first factorize out the coefficient

*Q(x) is just the rest of the polynomial

Remainder Theorem

• The remainder theorem is used to find the remainder when dividing a polynomial function by a linear function

• When any polynomial P(x) is divided by any linear function (x - k) the value of the remainder R is given by P(k) = R

  • Note, when P(k) = 0 then (x - k) is a factor of P(x)

Factor Theorem Tip

If asked to find integer solutions to a polynomial, then only consider factors of the constant term

• E.g: The integer solutions to the polynomial - x² - 5x + 6

• Consider the factors of 6: ±1,±2,±3,±6

• They could be the solutions to the polynomial

Using the Remainder Theorem

If the linear function has a coefficient of ‘x’, then first factorize out the coefficient