# In factorizing the expression a^4+a^2*b^2+b^4, we find that A. (a^2–b^2) is a factor. B. (a^2+ b^2) is a factor. C. (a^2− ab–b^2) is a factor. D. (a^2− ab+ b^2) is a factor. E. it...

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

A. (*a^2*–*b^2*) is a factor.

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

C. (*a^2*− *ab*–*b^2*) is a factor.

D. (*a^2*− *ab*+ *b^2*) is a factor.

E. it cannot be factorized.

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Expert Answers

jeew-m | Certified Educator

`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)*

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