x4 + y4 + x2y2 factorize?
Answers
Answer:
4(X+y4+2y2) this is ur answer
Answer:
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).