# How do you use the remainder theorem to see if the #n+5# is a factor of #n^5-25n^3-7n^2-37n-18#?

##### 2 Answers

#### Explanation:

Let

To see if

So

There is a remainder of

Therefore

Analyse the value of the polynomial for

#### Explanation:

Given:

#f(n) = n^5-25n^3-7n^2-37n-18#

The remainder theorem tells us that

Observe that all of the terms of

#f(n) = n^5-25n^3-7n^2-37n-18#

#color(white)(f(n)) = n(n^4-25n^2-7n-37)-18#

So if

#f(color(blue)(-5)) = (color(blue)(-5))((color(blue)(-5))^4-25(color(blue)(-5))^2-7(color(blue)(-5))-37)-18#

#color(white)(f(color(white)(-5))) = (color(blue)(-5))((color(blue)(-5))^4-25(color(blue)(-5))^2-7(color(blue)(-5))-37)-5*4+2#

#color(white)(f(color(white)(-5))) = 5k+2" "# for some integer#k#

#color(white)(f(color(white)(-5))) != 0#

Since