• Factor Theorem

Factor Theorem

If a polynomial $P(x)$ is divided by $(x-k)$ and there is no remainder, then the value $(x-k)$ is a factor of $P(x)$

$$ \text{remainder} = f(x)_{x \to k} = 0$$

Example:

1. When 2x³-bx²+6x-3b is divided by (x+2), the remainder is zero. Determine the value of b.
Solution:
$$ \text{Let } P(x) = 2x^3-bx^2+6x-3b$$ $$\begin{align} \text{Divisor: } & x+2 \\ & x + 2 = 0 \\ & x = -2 \\ \end{align}$$ @$x = -2$, (substitute to the polynomial)
$$\begin{align} remainder = P(x) = P(-2) & = 0\\ 2(-2)^3 - b(-2)^2 + 6(-2) - 3b & = 0\\ -28 - 7b & = 0 \\ b & = -4 \\ \end{align}$$ Therefore, $b = -4$.


2. What is the value of 'k' if (x+4) is a factor of x³+2x²-7x+k
Solution:
$$\text{Let }P(x) = x^3 + 2x^2 - 7x + k = 0 $$ $$\begin{align} \text{Divisor: } & x + 4 \\ & x + 4 = 0 \\ & x = -4\\ \end{align}$$ @$x = -4$, (substitute to the polynomial) $$\begin{align} remainder = P(x) = P(-4) & = 0 \\ (-4)^3 + 2(-4)^2 - 7(-4) + k & = 0\\ -4 + k & = 0 \\ k & = 4\\ \end{align}$$ Therefore, $k = 4$.