Can you show the following?

Show that x+3 is a factor of P(x)=x^4+3x^3-9x^2-27x

2 Answers
Mar 26, 2018

color(red)( x+3 color(red)("is a factor of" color(red)(x^4+3x^3-9x^2-27x

Explanation:

If x+3 is a factor of x^4+3x^3-9x^2-27x,
then x+3=0

therefore x=-3

(-3)^4+3(-3)^3-9(-3)^2-27(-3)

=cancel81-cancel81+cancel81-cancel81

=0

therefore color(red)( x+3 color(red)("is a factor of" color(red)(x^4+3x^3-9x^2-27x

~Hope this helps! :)

Mar 26, 2018

Using the remainder theorem:

f(x)=g(x)(x-a)+r

Where r is the remainder and g(x) is the quotient.

We know from this that if we can make (x-a)=0, then this gives the remainder. If P(x) is divisible by (x+3) then the remainder will be zero:

P(-3)=g(x)(-3+3)+0

P(-3)=0

(-3)^4+3(-3)^3-9(-3)^2-27(-3)=0

So (x+3) is a factor of P(x)