Remainder and Factor Theorem

The remainder and factor theorem states that when a polynomial f(x) is divided by (x-a), the remainder is f(a), and when f(a) is 0, (x-a) is a factor of f(x).

When dividing a polynomial, f(x) by another polynomial, g(x), we get a quotient polynomial q(x) and a remainder, r(x) if g(x) isn't a factor of f(x). This can be written as an equation: \(f(x)=g(x)q(x)+r(x)\).

Note that r(x) will always be zero or a polynomial of a lower degree than g(x).

Hence, when we divide a polynomial f(x) by (x-a) the remainder will always be a constant. This can be written as \(f(x)=(x-a)q(x)+r\). This means that \(f(a)=(a-a)q(x)+r=r\), establishing the remainder theorem.

When f(a)=0 this means that \(f(x)=(x-a)q(x) + 0\), showing that (x-a) is a factor of f(x).

Example

Given than \(f(x)=2x^3-hx^2+2x+12\) and (x-3) is a factor of f(x), find the value of h.

Solution

By the remainder and factor theorem, f(3)=0.

Also, \(f(3)=2(3)^3-h(3)^2+2(3)+12=-9h+72\). This means that \(-9h+72=0\) therefore h=8.

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