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Synthetic division And the Remainder theorem

One factor of f(x)=4x^3-4x^2-16x+16 is (x – 2). What are all the roots of the function? Use the Remainder Theorem.
A) x = 1, x = 2, or x = 4
B) x = –2, x = 1, or x = 2
C) x = 2, x = 4, or x = 16
D) x = –16, x = 2, or x = 16

Respuesta :

LammettHash
The remainder theorem says that if [tex]x-c[/tex] is a factor of a polynomial [tex]p(x)[/tex], then the remainder upon dividing [tex]\dfrac{p(x)}{x-c}[/tex] is 0 so that there exists a lower degree polynomial [tex]q(x)[/tex] as the quotient:

[tex]\dfrac{p(x)}{x-c}=q(x)[/tex]

Using the fact that [tex]x-2[/tex] is a factor, you can find a quadratic [tex]q(x)[/tex] which is easy to factorize further.

Synthetic division yields

[tex]q(x)=4x^2+4x-8[/tex]

which can be factored further as

[tex]4(x^2+x-2)=4(x+2)(x-1)[/tex]

So,

[tex]f(x)=4x^3-4x^2-16x+16=4(x-2)(x+2)(x-1)[/tex]

The roots are then [tex]x=-2,1,2[/tex].


Ver imagen LammettHash
ljmecham2001

Answer:

The answer to this question is "B.) x = -2, x = 1, or x = 2"