Question 4: Factorize a 4 + 3a 2 b 2 + 4b 4

a 4 + a 2 b 2 + b 4 or a 4 + 4b 4 type expressions

Question 4: Factorize $a^4+3a^2b^2+4b^4$

Factorization rules —

Solution:

Given expression is ;

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

This can be written as;

$=a^4+3a^2b^2+4b^4 +a^2b^2-a^2b^2$ (adding and subtracting $a^2b^2$ )

$=a^4+4a^2b^2+4b^4 -a^2b^2$

Arranging to form $a^2+2(a)(b)+b^2$ form for first three terms;

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

Taking $(a+b)^2$ form for fist three terms in this way;

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

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

This is now $a^2-b^2$ form- writing in brackets;

$=[(a^2+2b^2)^2 -(ab)^2 ]$

Applying $(a+b)(a-b)$ form;

$=[(a^2+2b^2)+ab ] [(a^2+2b^2)-ab ]$

$=[a^2+2b^2+ab ] [a^2+2b^2-ab ]$

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