Factor completely $x ^ { \frac { 1 } { 2 } } - 10 x ^ { \frac { 1 } { 4 } } - 24$ by comparing it to $x ^ { 2 } - 10 x - 24$

Question

Quizplus · Accepted Answer

Let us substitute $y=x^{\frac{1}{4}}$. Then we have:

$$y^2 - 10y -24 = (y-12)(y+2)$$

Substituting back for $x$ and using the above factorization, we get:

\begin{align*}
x ^ { \frac { 1 } { 2 } } - 10 x ^ { \frac { 1 } { 4 } } - 24 &= (x ^ { \frac { 1 } { 4 } }) ^ 2 - 10(x ^ { \frac { 1 } { 4 } }) - 24 \
&= (x^{\frac{1}{4}}-12)(x^{\frac{1}{4}}+2) \
&= \left(x ^ { \frac { 1 } { 4 } } + 2 \left( x ^ { \frac { 1 } { 4 } } - 12 ight) ight)
\end{align*}

Therefore, the answer is $\boxed{	extbf{(C)}}$.

Factor Completely x12−10x14−24x ^ { \frac { 1 } { 2 } } - 10 x ^ { \frac { 1 } { 4 } } - 24x21​−10x41​−24

Factor Completely $x ^ { \frac { 1 } { 2 } } - 10 x ^ { \frac { 1 } { 4 } } - 24$