MHF4U
Remainder Theorem and Factor Theorem
Use long division to divide 𝑃(𝑥) = 𝑥 3 + 6𝑥 2 + 2𝑥 − 4 by 𝑥 − 1 and then by 𝑥 + 2.
What is the remainder?
Find the value of 𝑃(1) and 𝑃(−2).
What do you notice?
Remainder Theorem: When a polynomial function 𝑃(𝑥) is divided by 𝑥 − 𝑏, the remainder is _______
and when it is divided by 𝑎𝑥 − 𝑏, the remainder is _______, where 𝑎 and 𝑏 are integers and 𝑎 ≠ 0.
*This theorem allows us to determine the remainder without performing actual division.
Example 1 Find the reminder when 3𝑥 2 − 2𝑥 2 + 𝑥 − 5 is divided by 𝑥 + 2.
Example 2 Find the remainder exactly when 𝑥 3 − 4𝑥 2 + 2𝑥 − 6 is divided by 2𝑥 − 1.
Example 3 When 𝑥 3 − 𝑘𝑥 2 + 17𝑥 + 6 is divided by 𝑥 − 3, the remainder is 12. Find 𝑘.
How do we factor polynomials of degree 3 or higher?
Example: 𝑓(𝑥) = 𝑥 3 + 8𝑥 2 + 19𝑥 + 12
Factor theorem: (𝑥 − 𝑏) is a factor of a polynomial 𝑃(𝑥) iff (if and only if) ______________. Similarly,
(𝑎𝑥 − 𝑏) is a factor of 𝑃(𝑥) iff __________
Example 4
Is (𝑥 − 2) a factor of 𝑥 3 − 7𝑥 2 + 9𝑥 + 2?
Example 5
a) Find a factor of 𝑥 3 + 8𝑥 2 + 19𝑥 + 12
b) Use the factor from a) to fully factor the polynomial 𝑥 3 + 8𝑥 2 + 19𝑥 + 12
What happens when we have a leading coefficient that is not 1?
𝑏
𝑎
Rational zero theorem: If 𝑃(𝑥) is a polynomial function with integer coefficients and 𝑥 = is a zero of
𝑃(𝑥), where 𝑎, 𝑏 are integers and 𝑎 ≠ 0 then
𝑏 is a factor of the constant term
𝑎 is a factor of the leading coefficient
𝑎𝑥 − 𝑏 is a factor of 𝑃(𝑥)
Example 6
a) List possible factors of 5𝑥 3 + 3𝑥 2 − 12𝑥 + 4
b) Find the factor of 5𝑥 3 + 3𝑥 2 − 12𝑥 + 4
c) Factor 5𝑥 3 + 3𝑥 2 − 12𝑥 + 4
Example 7
Factor
a) 𝑥 3 − 𝑥 2 − 14𝑥 + 24
b) 𝑥 3 − 4𝑥 2 + 𝑥 + 6