# Factorize the algebraic expression $a^2+2ab+b^2-16$.

Given:

The given expression is $a^2+2ab+b^2-16$.

To do:

We have to factorize the algebraic expression $a^2+2ab+b^2-16$.

Solution:

Factorizing algebraic expressions:

Factorizing an algebraic expression implies writing the expression as a product of two or more factors. Factorization is the reverse of distribution.

An algebraic expression is factored completely when it is written as a product of prime factors.

$a^2+2ab+b^2-16$ can be written as,

$a^2+2ab+b^2-16=a^2+2(a)(b)+(b)^2-16$             [Since $a^2=(a)^2, b^2=(b)^2$ and $2ab=2(a)(b)$]

Here, we can observe that the given expression is of the form $m^2+2mn+n^2$. So, by using the formula $(m+n)^2=m^2+2mn+n^2$, we can factorize the given expression.

Here,

$m=a$ and $n=b$

Therefore,

$a^2+2ab+b^2-16=(a)^2+2(a)(b)+(b)^2-16$

$a^2+2ab+b^2-16=(a+b)^2-16$

Now,

$(a+b)^2-16$ can be written as,

$(a+b)^2-16=(a+b)^2-4^2$        [Since $16=4^2$]

Using  the formula $a^2-b^2=(a+b)(a-b)$, we can factorize $(a+b)^2-4^2$ as,

$(a+b)^2-16=(a+b)^2-4^2$

$(a+b)^2-16=(a+b+4)(a+b-4)$

Hence, the given expression can be factorized as $(a+b+4)(a+b-4)$.

Updated on: 10-Apr-2023

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