Find the EXACT Zeros of the polynomial function f(x)= x^3 - 8x^2 - 7x + 56 by factoring.

Question

Find the EXACT Zeros of the polynomial function f(x)= x^3 - 8x^2 - 7x + 56 by factoring.

The Zeros are:____________. Rounded numbers will not be accepted. Use a comma to separate the Zeros if the polynomial has more than one.

Write the factored form of the polynomial function.

f(x)=_______________.

Answer 1

x^3 - 8x^2 -7x +56

by decartes rule, 2 possible possible real, 1 possible negative real solutions

x=8 is one positive real solution, (x-8) is one factor

use synthetic or long division to find the quadratic: x^2-7

x^2-8x^2-7x +56 = (x-8)(x^2-7) = (x-8)(x-sqr7)(x+sqr7)

zeros are 8, sqr7, -sqr7

2 real positive solutions and 1 negative real solution

use a graphing calculator and where the graph intersects the x axis are the zeros

or notice 56=7x8, so factor out 8 from the last 2 terms and factor out x^2 from first 2 terms

x^3-8x^2-7x +56

=x^2(x-8) -7(x-8)

= (x-8)(x^2-7)

set each factor = 0 and solve for x

x=8

and

x^2-7=0

x^2=7

x = sqr7 and -sqr7