Answer:
The roots of f(x) are x=2, 1, -2
Step-by-step explanation:
Given polynomial is [tex]f(x)=4x^{3}-4x^{2}-16x+16[/tex]
One factor of above polynomial is (x – 2). we have to find all the roots of above polynomial. By applying synthetic division we find the quotient of above polynomial when divided by (x-2).
We get the coefficient [tex]4x^{2}+4x-8[/tex]
Hence, by remainder theorem,
[tex]f(x)=4x^{3}-4x^{2}-16x+16=(x-2)(4x^{2}+4x-8)+0[/tex]
=[tex](x-2)4(x^{2}+x-2)[/tex]
=[tex](x-2)(x^{2}+2x-x-2)4[/tex]
=[tex](x-2)(x(x+2)-1(x+2))4[/tex]
=[tex](x-2)(x-1)(x+2)4[/tex]
To find roots, f(x)=0
⇒[tex](x-2)(x-1)(x+2)4=0[/tex]
⇒ x=2, 1, -2
