Factor the polynomials completely.

$$
16 x^4-200 x^2+625
$$

Question

Factor the polynomials completely.

$$
16 x^4-200 x^2+625
$$

Quizlet Admin · Accepted Answer

We have
\begin{flalign*}
&16x^4 - 200x^2 + 625\
&= 4^2x^{2 \cdot 2} - 200x^2 + 625\
&= 4^2(x^2)^2 - 200x^2 + 625\,\,\,\,\mbox{[by the power rule of exponents]}\
&= (4x^2)^2 - 200x^2 + 625\,\,\,\,\mbox{[by the power of a product rule of exponents]}\
&= (4x^2)^2 - 2 \cdot 4x^2 \cdot 25 + 25^2\
&= (4x^2 - 25)^2\,\,\,\,\mbox{[by the formula $a^2 - 2ab + b^2 = (a - b)^2$]}\
&= (2^2x^2 - 5^2)^2\
&= [(2x)^2 - 5^2]^2\,\,\,\,\mbox{[by the power of a product rule of exponents]}\
&= [(2x + 5)(2x - 5)]^2\,\,\,\,\mbox{[by the formula $a^2 - b^2 = (a + b)(a - b)$]}\
&= (2x + 5)^2(2x - 5)^2\,\,\,\,\mbox{[by the power of a product rule of exponents]}\
\end{flalign*} 
Therefore factoring the given polynomial, we get $(2x + 5)^2(2x - 5)^2.$

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