Question

Given one factor of the polynomial, factor the polynomial completely.

Answer 1

Use synthetic division to solve these problems

5 | 1    -4   -7    10

  |  v    5    5   -10

     1     1    -2    0

This leads to x^2+x-2          

Next we find what adds to 1 and multiplies to -2

The factors of f(x) are 2 and -1, written out as (x+2) and (x-1)

-1/2 | 2     3     -23    -12

      | v     -1       -1      12

        2      2    -24     0

This leads to 2x^2+2x-24

Next, multiply the 2 from the a value (since a is the value with x^2 and c is the value without any x) with -24, which is -48

Now find what adds to 2 and multiplies to -48

The values for this are 8 and -6, but keep in mind, when multiplying the value of a by c, you now have to add the values in place of what you tried adding it to, this time in the problem we found using synthetic division, in this case the 2 in the b value (the middle of the equation, or the value with only one x)

2x^2+8x-6x-24

When group factoring, you'd get 2x(x+4) -6 (x+4)

So, the factors of g(x) are (x+4) and (2x-6)