Question

Complete the steps to factor the polynomial. One root of f(x) = x³ + x² - 22x - 40 is 5. If 5 is a root of the function, then is a factor.

Answer 1

Then x-5 is a factor 

Answer 2

If one root of a cubic is known (and verified), then we can do a synthetic division and factor the remaining quadratic expression.

f(x)=x^3+x^2-22x-40
f(5)=125+25-22(5)-40=0  so x-5 is a factor

Now do the synthetic division

  5  |  +1 +1 -22 -40
      --------------------
           1  6    8    0

The resulting quadratic is x^2+6x+8 which can be factored into
(x+2)(x+4)

The complete factorization is therefore
f(x)=x³ + x² - 22x - 40 = (x-5)(x+2)(x+4)