Skill: Use the remainder theorem

The remainder theorem says:

When a polynomial $f(x)$ is divided by $x-a$, the remainder is $f(a)$.

So, instead of performing long division, substitute $a$ into the polynomial.

Example

Find the remainder when

$$f(x)=2x^3-3x^2+4x-5$$

is divided by $x-2$.

Step 1: Identify $a$

The divisor is $x-2$, which has the form $x-a$. Therefore,

$$a=2$$

Step 2: Evaluate $f(2)$

Substitute $2$ for $x$:

$$f(2)=2(2)^3-3(2)^2+4(2)-5$$

Simplify:

$$f(2)=2(8)-3(4)+8-5$$ $$f(2)=16-12+8-5=7$$

Step 3: State the remainder

The remainder is

$$\boxed{7}$$

Therefore, dividing $2x^3-3x^2+4x-5$ by $x-2$ leaves a remainder of $7$.

If $f(a)=0$, then the remainder is zero, which means $x-a$ is a factor of $f(x)$.