# 3.05 Remainder and factor theorems

## Interactive practice questions

Christa wants to test whether various linear expressions divide exactly into $P\left(x\right)$P(x), or whether they leave a remainder. For each linear expression below, state the value of $x$x that needs to be substituted into $P\left(x\right)$P(x) to find the remainder.

a

$x+3$x+3

b

$8-x$8x

c

$5+4x$5+4x

d

$6-x$6x

Easy
Approx 2 minutes

Fill in the gap to make the statement true.

If $P\left(x\right)=-2x^4+7x^3-3x^2-6x-5$P(x)=2x4+7x33x26x5 and $A\left(x\right)=x+2$A(x)=x+2. Use the remainder theorem to find the remainder when $P\left(x\right)$P(x) is divided by $A\left(x\right)$A(x).

Using the remainder theorem, find the remainder when $P\left(x\right)=-4x^4+6x^3+4x^2-7x+7$P(x)=4x4+6x3+4x27x+7 is divided by $A\left(x\right)=3x-1$A(x)=3x1.

### Outcomes

#### MGSE9-12.A.APR.2

Know and apply the Remainder Theorem: For a polynomial p(x) and a number a, the remainder on division by x â€“ a is p(a), so p= 0 if and only if (x â€“ is a factor of p(x).