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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 $(a+b+4)(a+b-4)$.
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