Factorize $5a(x^2-y^2)$ $+$ $35b(x^2-y^2)$

The polynomial contains two terms $5a(x^2-y^2)$ and $35b(x^2-y^2)$, and let’s express both expressions as a product of primary level factors.

$(1).\,\,$ $5a(x^2-y^2)$ $\,=\,$ $5 \times a \times (x^2-y^2)$

$(2).\,\,$ $35b(x^2-y^2)$ $\,=\,$ $5 \times 7 \times b \times (x^2-y^2)$

So, let’s identify the commonly appearing factors in the terms $5a(x^2-y^2)$ and $35b(x^2-y^2)$ by comparing their primary level factors.

$(1).\,\,$ $5a(x^2-y^2)$ $\,=\,$ $\underline{5} \times a \times \underline{(x^2-y^2)}$

$(2).\,\,$ $35b(x^2-y^2)$ $\,=\,$ $\underline{5} \times 7 \times b \times \underline{(x^2-y^2)}$

In the above expressions, $5$ and $x^2-y^2$ are identified as common factors. Now, write the common factors closely in the product as per the commutative property.

$(1).\,\,$ $5a(x^2-y^2)$ $\,=\,$ $5 \times (x^2-y^2) \times a$

$(2).\,\,$ $35b(x^2-y^2)$ $\,=\,$ $5 \times (x^2-y^2) \times 7 \times b$

Finally, multiply the common factors $5$ and $x^2-y^2$, and their product is $5(x^2-y^2)$, which is called the greatest or highest common factor (HCF). Similarly, multiply the other factors separately in both expressions.

$(1).\,\,$ $5a(x^2-y^2)$ $\,=\,$ $5(x^2-y^2) \times a$

$(2).\,\,$ $35b(x^2-y^2)$ $\,=\,$ $5(x^2-y^2) \times 7b$

Now, the given polynomial $5a$ times $x^2$ minus $y^2$ plus $35b$ times $x^2$ minus $y^2$ can be written in the greatest common factor form as follows.

$\implies$ $5a(x^2-y^2)$ $+$ $35b(x^2-y^2)$ $\,=\,$ $5(x^2-y^2) \times a$ $+$ $5(x^2-y^2) \times 7b$

Now, let’s focus on simplifying the algebraic expression $5(x^2-y^2) \times a$ $+$ $5(x^2-y^2) \times 7b$. In this expression, the GCF $5(x^2-y^2)$ can be taken out, from the terms and it can be done by using the distributive property.

$\,\,=\,\,$ $5(x^2-y^2) \times (a+7b)$

$\,\,=\,\,$ $5(x^2-y^2)(a+7b)$