Factor & Remainder Theorem (DP IB Analysis & Approaches (AA)): Revision Note

Factor Theorem

What is the 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

• 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)

How do I use the factor theorem?

• Consider the polynomial function P(x) = a_nxⁿ + a_n-1x^n-1 + … + a₁x + a₀ and (x - k) is a factor

  • Then, due to the factor theorem P(k) = a_nkⁿ + a_n-1k^n-1 + … + a₁k + a₀ = 0

  • , where Q(x) is a polynomial that is a factor of P(x)

  • Hence,  , where Q(x) is another factor of P(x)

• If the linear factor has a coefficient of x then you must first factorise out the coefficient

  • If the linear factor is 

Remainder Theorem

What is the remainder theorem? 

• The remainder theorem is used to find the remainder when we divide 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)

How do I use the remainder theorem?

• Consider the polynomial function P(x) = a_nxⁿ + a_n-1x^n-1 + … + a₁x + a₀ and the linear function (x - k) 

  • Then, due to the remainder theorem P(k) = a_nkⁿ + a_n-1k^n-1 + … + a₁k + a₀ = R

  • , where Q(x) is a polynomial

  • Hence,  , where R is the remainder

• If the linear function has a coefficient of x then you must first factorise out the coefficient

  • If the linear function is