Question

One factor of $x^4-(x-z{)}^4$ is __________.

• A. $2x+z$
• B. $x+2z$
• C. $2x-z$
• D. $x-2z$

Answer 1

The correct option is C $2x-z$

Given, $x^4-(x-z{)}^4$

We know, $a^2-b^2=(a+b)(a-b)$ $(\because \text{Identity})$

Here, $a=x^2$ and $b=(x-z{)}^2$

$\Rightarrow x^4-(x-z{)}^4=(x^2+(x-z{)}^2)(x^2-(x-z{)}^2)$

We can apply same Identity $m^2-n^2=(m+n)(m-n)$ on the $Bold$ factor.

Now, $m=x$ and $n=(x-z)$

$\Rightarrow x^2-(x-z{)}^2=(x+(x-z)\left)\right(x-(x-z))$ $=(2x-z)\left(z\right)$

Therefore, $x^4-(x-z{)}^4=(z\left)\right(2x-z\left)\right(x^2+(x-z{)}^2)$

Thus one of the factors is $2x-z$.