Factor the expression using the two different techniques listed for parts 1(a) and 1(b). (a) factor the given expression using the gcf monomial. show all of your work. (b) factor the given expression using the difference of squares. show all of your work.

9a^4 b^10(2 +3a^6 b^5)  (2 -3a^6 b^5)

see work below

Step-by-step explanation:

36a^4b^10 - 81a^16b^20

A)  find the GCF

36a^4b^10 = 4*9 a^4b^10  = 2*2*3*3 a*a*a*a*b*b*b*b*b*b*b*b*b*b

81a^16b^20= 9*9a^16b^20= 3*3*3*3* a*a*a*a*a*a*a*a*a*a*a*a*a*a*a*a                *b*b*b*b*b*b*b*b*b*b*b*b*b*b*b*b*b*b*b*b

The terms that appear in both terms is the GCF.  The terms that remain are inside the parentheses.

The GCF is 3*3 a*a*a*a*b*b*b*b*b*b*b*b*b*b

36a^4b^10 - 81a^16b^20 = 3*3 a*a*a*a*b*b*b*b*b*b*b*b*b*b *

( 2*2 - 3*3a*a*a*a*a*a*a*a*a*a*a*a b*b*b*b*b*b*b*b*b*b)

Combining like terms

36a^4b^10 - 81a^16b^20 = 9a^4b^10(4-9a^12b^10)

The expression inside the parenthesis can be factored using the difference of squares

let x^2 =4   x =2  

y^2 = 9a^12 b^10   y = 3a^6b^5

(x^2 -y^2) = (x+y)(x-y)

9a^4b^10(4-9a^12b^10) = 9a^4b^10 ( 2+3a^6b^5) ( 2-3a^6b^5)

b) difference of squares  a^2 – b^2 = (a + b)(a – b)

let a^2 = 36a^4b^10

so a = 6a^2b^5

b^2 = 81a^16b^20

b = 9a^8 b^10

a^2 – b^2 = (a + b)(a – b)

36a^4b^10 - 81a^16b^20 = (6a^2b^5 +9a^8 b^10) (6a^2b^5 -9a^8 b^10)

We can factor a 3 a^2 b^5 out of the first term

3 a^2 b^5 (2 +3a^6 b^5) (6a^2b^5 -9a^8 b^10)

3 a^2 b^5 (2 +3a^6 b^5) 3 a^2 b^5 (2 -3a^6 b^5)

Multiply the terms outside the parentheses together

9a^4 b^10(2 +3a^6 b^5)  (2 -3a^6 b^5)

same ans for a n b

9a^4b^10(2+6a^6b^5)(2-6a^6b^5)

Step-by-step explanation:

(a) 36a^4b^10 - 81a^16b^20 by GCF

compare const terms: 36 n 81, GCF is 9x4 n 9x9

compare terms w a: a^4 n a^16, GCF is a^4 n a^4xa^12

compare terms w b: b^10 n b^20, GCF is b^10 n b^10xb^10

combining GCFs, 36a^4b^10 - 81a^16b^20

=9a^4b^10(4 - 9a^12b^10)

note 4=2^2 n 9a^12b^10=(3a^6b^5)^2

using difference of squares eqn A^2 - B^2 = (A+B)(A-B)

4 - 9a^12b^10 = (2+3a^6b^5)(2-3a^6b^5)

substitute back

36a^4b^10 - 81a^16b^20

=9a^4b^10(2+3a^6b^5)(2-3a^6b^5)

(b) using difference of squares eqn A^2 - B^2 = (A+B)(A-B)

36a^4b^10=(6a^2b^5)^2

81a^16b^20=(9a^8b^10)^2

36a^4b^10 - 81a^16b^20

=(6a^2b^5+9a^8b^10)(6a^2b^5-9a^8b^10)

=3a^2b^5(2+6a^6b^5)3a^2b^5(2-6a^6b^5)

=9a^4b^10(2+6a^6b^5)(2-6a^6b^5)

same as a