Once you know that the binomial is a factor of the polynomial, how do you factor it completely?

Question

"Show that the binomial is a factor of the polynomial. Then factor the polynomial completely. "

The last part is what I am confused about. I understand that once the remainder is zero it means it is a factor, but I do not know how to take that information and factor it.

Example Problems:

x⁴ - 6x³ - 8x + 48; (x - 6)

or

x⁴ + 4x³ - 64x - 256; (x + 4)

Answer 1

Hi Makaylee,

To see if the binomial is a factor of the polynomial, you would divide the polynomial by the binomial (ie., Long division). If there is no remainder, then the binomial is the factor of the polynomial.

That is, for example 1:

_________________

x-6 ) X^4 - 6X^3 - 8X + 48

= X^3 - 8

Since no remainder, X -6 is a factor of the polynomial. And X^3 - 8 is a difference of cubes Which can be factored further to:

a³ – b³ = (a – b)(a² + ab + b²)

OR

X^3 - 8 = (X -3)(X^2 + 2X + 4)

Final factoring, putting it all together ...

(X -6) (X -3)(X^2 + 2X + 4)