Homework Help

In factorizing the expression a^4+a^2*b^2+b^4, we find that A. (a^2–b^2) is a...

user profile pic

christiano-cr7 | Salutatorian

Posted September 2, 2013 at 3:22 AM via web

dislike 1 like

In factorizing the expression a^4+a^2*b^2+b^4, we find that

A. (a^2b^2) is a factor.

B. (a^2+ b^2) is a factor.

C. (a^2abb^2) is a factor.

D. (a^2ab+ b^2) is a factor.

E. it cannot be factorized.

1 Answer | Add Yours

user profile pic

jeew-m | College Teacher | (Level 1) Educator Emeritus

Posted September 2, 2013 at 3:40 AM (Answer #1)

dislike 1 like

`a^4+a^2xxb^2+b^4`

Let;

`x = a^2`

`y = b^2`

So we can rewrite the expression as;

`x^2+xy+y^2`

`f(x) = x^2+xy+y^2`

`f(x) = x^2+2xy-xy+y^2`

`f(x) = (x^2+2xy+y^2)-xy`

We know that `x^2+2xy+y^2 = (x+y)^2`

`f(x) = (x+y)^2-xy`  

Substituting original values of x and y yields;

`(x+y)^2-xy = (a^2+b^2)^2-a^2b^2`

Therefore;

`a^4+a^2xxb^2+b^4`

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

`= (a^2+b^2-ab)(a^2+b^2+ab)`

`= (a^2-ab+b^2)(a^2+ab+b^2)`

So the correct answer is D)

Sources:

Join to answer this question

Join a community of thousands of dedicated teachers and students.

Join eNotes