Question

Which of the following is a factor of $2x^4+1-3x^2$

• A. $1+2x^2$
• B. $4+x^2$
• C. $2x^2-1$
• D. $3+x^2$

Answer 1

Answer: (C)

$2x^2-1$

In the equation $2x^4+1-3x^2$, rewrite $x^4$ as $(x^2{)}^2$ as follows:

$2(x^2{)}^2+1-3x^2$

Let $u=x^2$, then the above equation becomes $2u^2-3u+1$, and we factorize the equation as shown below:

$2u^2-3u+1=2u^2-2u-u+1=2u(u-1)-1(u-1)=(2u-1)(u-1)=(2x^2-1)(x^2-1)(\because u=x^2)=(2x^2-1)(x^2-1^2)=(2x^2-1)(x+1)(x-1)(\because a^2-b^2=(a+b\left)\right(a-b\left)\right)$

Hence, one of the factor of $2x^4+1-3x^2$ is $2x^2-1$.