What are all the zeroes of #f(x) = 2x^3 - 2x^2 - 8x + 8#?

1 Answer
Dec 13, 2015

Answer:

#x=-2,1,2#

Explanation:

Factor the polynomial.

#f(x)=2(x^3-x^2-4x+4)#

Factor by grouping.

#f(x)=2(x^2(x-1)-4(x-1))#

#f(x)=2(x^2-4)(x-1)#

Recognize that #x^2-4# is a difference of squares.

#f(x)=2(x+2)(x-2)(x-1)#

Now, to find the zeros of a function, find the times when #f(x)=0#.

#0=2(x+2)(x-2)(x-1)#

When there is a product of things that equal #0#, at least one of the terms MUST equal #0#. To find when this happens, set each non-constant term equal to #0#.

#x+2=0#
#color(blue)(x=-2)#

#x-2=0#
#color(blue)(x=2)#

#x-1=0#
#color(blue)(x=1)#

A good way to double check is by checking a graph.
graph{2x^3-2x^2-8x+8 [-7.96, 12.04, -3.56, 6.44]}

The zeros, when the graph crosses the #x#-axis, occur at #x=-2,1,2#.