Question

Factor completely or state that the polynomial is prime.2x^4-2

Answer 1

2(x²+1)(x+1)(x-1)

1) Let's examine that polynomial to check whether that polynomial can be factored or not.

`2x^4-2`

2) Examining we can see that the coefficient of the x-term (2) and the constant (2), are the same. So we can find the GCD, between 2,2 and write it as a factor.

GCD (2, 2) = 2

Now we'll divide every term by the GCD, and write it inside the parentheses just like this

`\begin{gathered} 2x^4-2 \\ (2x^4)/(2)=x^4 \\ (-2)/(2)=-1 \\ \\ 2(x^4-1)^{} \end{gathered}`

2.2) Now let's proceed with the factorization of the binomial x^4-1

Remembering that (a+n)(a-b)=a²-b²

Notice that we are dealing with a 4th-degree polynomial, so we can write it as:

So we can rewrite it as

`2(x^4-1)=2(x^2+1)(x^2-1^2)\text{ =}2(x^2+1)(x+1)(x-1)`

3) So our polynomial is not prime, for it can be reduced. And it can be written as 2(x²+1)(x+1)(x-1) in its simplest form.