# Factorize the following algebraic expressions:(i) $a^2+4ab+3b^2$(ii) $96-4x-x^2$(iii) $a^4+3a^2+4$

Given:

The given expressions are:

(i) $a^2+4ab+3b^2$.

(ii) $96-4x-x^2$

(iii) $a^4+3a^2+4$

To do:

We have to factorize the given algebraic expressions.

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.

(i) The given expression is $a^2+4ab+3b^2$.

$a^2+4ab+3b^2$ can be written as,

By splitting and grouping the terms we can factorize the given expression.

$a^2+4ab+3b^2=a^2+ab+3ab+3b^2$                         [Since $4ab=ab+3ab$]

Therefore,

$a^2+4ab+3b^2=a^2+ab+3ab+3b^2$

$a^2+4ab+3b^2=a(a+b)+3b(a+b)$

$a^2+4ab+3b^2=(a+b)(a+3b)$

Hence, the given expression can be factorized as $(a+b)(a+3b)$.

(ii) The given expression is $96-4x-x^2$.

By splitting and grouping the terms we can factorize the given expression.

$96-4x-x^2$ can be written as,

$96-4x-x^2=-(x^2+4x-96)$

$96-4x-x^2=-(x^2+12x-8x-96)$                       (Since $4x=12x-8x$)

Therefore,

$96-4x-x^2=-(x^2+12x-8x-96)$

$96-4x-x^2=-[x(x+12)-8(x+12)]$

$96-4x-x^2=-(x+12)(x-8)$

$96-4x-x^2=(x+12)(8-x)$                         [Since $-(x-8)=(-x+8)=(8-x)$]

Hence, the given expression can be factorized as $(x+12)(8-x)$.

(iii) The given expression is $a^4+3a^2+4$.

By splitting and grouping the terms we can factorize the given expression.

$a^4+3a^2+4$ can be written as,

$a^4+3a^2+4=a^4+4a^2-a^2+4$                       (Since $3a^2=4a^2-a^2$)

$a^4+3a^2+4=(a^2)^2+2(a^2)(2)+2^2-a^2$

$a^4+3a^2+4=(a^2+2)^2-a^2$                          [Since $(m+n)^2=m^2+2mn+n^2$]

Therefore, by using the formula $m^2-n^2=(m+n)(m-n)$, we can factorize the given expression.

$a^4+3a^2+4=(a^2+2)^2-a^2$

$a^4+3a^2+4=(a^2+2+a)(a^2+2-a)$

Hence, the given expression can be factorized as $(a^2+a+2)(a^2-a+2)$.

Updated on: 10-Apr-2023

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