Question

question 9 (5 points)
factor the following using the difference of squares technique:
a) $144x^{10}-25x^{26}$
b) $675x^{23}y^{7}-192xy^{9}$

Explanation:

Step1: Factor out common factors for part a

First, factor out $x^{10}$ from $144x^{10}-25x^{26}$. We get $x^{10}(144 - 25x^{16})$. Then, rewrite $144$ as $12^{2}$ and $25x^{16}$ as $(5x^{8})^{2}$.
$x^{10}(12^{2}-(5x^{8})^{2})$

Step2: Apply difference - of - squares formula for part a

Using the formula $a^{2}-b^{2}=(a + b)(a - b)$, where $a = 12$ and $b = 5x^{8}$, we have:
$x^{10}(12 + 5x^{8})(12 - 5x^{8})$

Step3: Factor out common factors for part b

Factor out $3xy^{7}$ from $675x^{23}y^{7}-192xy^{9}$. We get $3xy^{7}(225x^{22}-64y^{2})$. Then, rewrite $225x^{22}$ as $(15x^{11})^{2}$ and $64y^{2}$ as $(8y)^{2}$.
$3xy^{7}((15x^{11})^{2}-(8y)^{2})$

Step4: Apply difference - of - squares formula for part b

Using the formula $a^{2}-b^{2}=(a + b)(a - b)$, where $a = 15x^{11}$ and $b = 8y$, we have:
$3xy^{7}(15x^{11}+8y)(15x^{11}-8y)$

Answer:

a) $x^{10}(12 + 5x^{8})(12 - 5x^{8})$
b) $3xy^{7}(15x^{11}+8y)(15x^{11}-8y)$