MHB Is it necessary to expand the quantity before factoring the expression?

[mathdad]

Precalculus by David Cohen, 3rd Edition
Chapter 1, Section 1.3.
Question 29c.

Factor the expression.

4a^2b^2 - 9(ab + c)^2

Must I expand the quantity (ab + c) before factoring?

 

[kaliprasad]

Precalculus by David Cohen, 3rd Edition
Chapter 1, Section 1.3.
Question 29c.

Factor the expression.

4a^2b^2 - 9(ab + c)^2

Must I expand the quantity (ab + c) before factoring?

You need not

you get as $(2ab)^2 - (3(ab+c))^2$ difference of 2 squares and now you should be able to proceed

 

[mathdad]

(2ab)^2−(3(ab+c))^2

I will let 2ab = x and [3(ab + c)] = y.

x^2 - y^(2)

(x - y)(x + y)

Back-substitute now.

[(2ab - 3(ab + c)][(2ab + 3(ab + c)]

Correct?

 

[MarkFL]

(2ab)^2−(3(ab+c))^2

I will let 2ab = x and [3(ab + c)] = y.

x^2 - y^(2)

(x - y)(x + y)

Back-substitute now.

[(2ab - 3(ab + c)][(2ab + 3(ab + c)]

Correct?

Yes, although I would distribute within the factors and then combine like terms. :D

 

[mathdad]

I thought about distributing within the factors and combining like terms but the question is asking to factor not to simplify after factoring.

 

[MarkFL]

I thought about distributing within the factors and combining like terms but the question is asking to factor not to simplify after factoring.

Well, technically, further simplification is not required to complete the goal of factorization...you have indeed factored...but in my opinion, it's just good practice to simplify whenever possible.

 

[mathdad]

Well, technically, further simplification is not required to complete the goal of factorization...you have indeed factored...but in my opinion, it's just good practice to simplify whenever possible.

I agree.