Question

a) \(4x^4+1\)

b) \(4x^4+81\)

c) \(64x^4+y^4\)

d) \(x^8+4\)

e) \(x^4+x^2+1\)

f) \(x^7+x^5+1\)

Answer 1

a) = \(4x^4+4x^2+1\)

= \(\left(2x^2+1\right)^2\)

b) = \(4x^4+36x^2+81-36x^2\)

= \(\left(2x^2+9\right)^2\)

c) = \(64x^4+16x^2y^2+y^4-16x^2y^2\)

= \(\left(8x^2+y^2\right)^2\)

d) = \(x^8+4x^4+4-4x^4\)

= \(\left(x^4+2\right)^2\)

e) = \(\left(x^4+2x^2+1\right)-x^2\)

= \(\left(x^2+1\right)^2-x^2\)

= \(\left(x^2+1-x\right).\left(x^2+1+x\right)\)

f) = \(\left(x^7-x\right)+\left(x^5-x^2\right)+\left(x^2+x+1\right)\)

= \(x.\left(x^3-1\right).\left(x^3+1\right)+x^2.\left(x^3-1\right)+\left(x^2+x+1\right)\)

\(\left(x^2+x+1\right).\left(x-1\right).\left(x^4+x\right)+x^2.\left(x-1\right).\left(x^2+x+1\right)+\left(x^2+x+1\right)=\left(x^2+x+1\right).\left(x^5-x^4+x^3-1+1\right)\)

Answer 2

c/=64x^4+16x^2y^2+y^4-16x^2y^2

=(8x^2+y^2)^2-(4xy)^2

=(8x^2+y^2+4xy)(8x^2+y^2-4xy)