Question

Factor the expression using the two different techniques listed for Parts 1(a) and 1(b).

Answer 1

Answer:

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

Answer 2

Answer:

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)