Question

$256 {x}^{4} - {x}^{2} {y}^{2} + 49 {y}^{4}$
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Answer 1

Answer:

I hope it will help you

Step-by-step explanation:

Factorization of x4 + x2y2 + y4

What you should know before…

a2 – b2 = (a + b) (a – b) (1)

a3 + b3 = (a + b) (a2 - ab + b2) (2)

a3 – b3 = (a – b) (a2 + ab + b2) (3)

a2 + 2ab + b2 = (a + b)2 (4)

The Round-about Tour

Let us begin with the factorization of x6 – y6 in two ways :

(a) x6 – y6 = (x2)3 – (y2)3 = (x2 – y2)[(x2)2 + x2y2 +(y2)2], by (3)

= (x + y)(x – y)(x4 + x2y2 + y4), by (1)

(b) x6 – y6 = (x3)2 – (y3)2 = (x3 + y3)(x3 – y3), by (1)

= (x + y)(x2 – xy + y2)(x - y)(x2 +xy +y2), by (2) and (3)

Which of the above factorization is correct?

Of course, (b) is the complete factorization, (a) is not.

Comparing the results in (a) and (b), we can get:

x4 + x2y2 + y4 = (x2 + xy + y2)(x2 –xy + y2)

Further investigation

x4 + x2y2 + y4 = (x4 + 2x2y2 + y4) - x2y2 (note that one term is added and subtracted)

= (x2 + y2)2 – (xy)2, , by (4)

= [(x2 + y2) + xy] [(x2 + y2) – xy] , by (1)

= (x2 + xy + y2)(x2 –xy + y2)

Similar way

There are some factorization which use the same technique, here is one example:

x4 + 4 = (x4 + 4x2 + 4) – 4x2

= (x2 + 2)2 – (2x)2

= (x2 + 2 + 2x)(x2 + 2 – 2x)

= (x2 + 2x + 2)(x2 – 2x + 2).