Algebra Help "Special Factoring"?

Question

if you can help me with these problems, I'll probably be able to do the rest :) Thanks
I need to factor them

x^2y^2-25
x^5-100x^3
25x^2yz-40xyz+16yz
x^2y^2-81
64x^2z-49z
x^2y^2-16xy^2+64y^2

Answer 1

x^2y^2 – 25

use a² – b² = (a + b)(a – b)

(xy)² – (5)² = (xy + 5)(xy – 5)



x^5 – 100x^3

coefficients are 1 and 100 GCD or common is 1

x^5 and x^3 GCD or common is lowest exponent of variable x = x^3

x^5 – 100x^3

= x^3(x^2 – 100)

= x^3(x + 10)(x – 10)



25x^2yz-40xyz+16yz

= yz(25x^2 – 40x + 16)

= yz(5x – 4)²





x^2y^2-81

same as 1st



64x^2z-49z

same as 3rd



x^2y^2-16xy^2+64y^2

same as 3rd take y^2 common

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Answer 2

x^2y^2-25 = [(xy)^2 - 5^2] =(xy-5(xy+5)



for this: x^5-100x^3 , factor out x^3, ie. x^5-100x^3 = x^3(x^2-100),

but we know (x^2-100) = (x+10)(x=10)

So: x^5-100x^3 = x^3 (x+10)(x=10)



25x^2yz-40xyz+16yz



x^2y^2-81 = (xy)^2 - 9^2 =(xy+9)(xy-9)



64x^2z-49z, factor out z, then 64x^2z-49z = z(64x^2 - 49),

but (64x^2 - 49)=(8x)^2-7^2 = (8x+7(9x-7)

So:

64x^2z-49z = z(8x+7)(9x-7)



for this: x^2y^2-16xy^2+64y^2 , first recognize the common term w/c is y^2, and factor it out

then:

x^2y^2-16xy^2+64y^2 = y^2(x^2-16x+64), see that x^2-16x+64 is a perfect quadratic eqn

i.e. x^2-16x+64 = (x-8)^2

So:

x^2y^2-16xy^2+64y^2 = y^2(x-8)^2 = y^2(x-8)(x-8)



Please review.

Cheers!

Answer 3

x²y² - 25 ends up as the difference of two squares.

(xy)² - 5²

This factories to (xy - 5) ( xy + 5)

[ Recall a² - b² = (a - b) (a + b)]

The other problems are very much similar. Wish you work them out yourself.

Good Luck !