Question

Create equivalent expressions by factoring
Problem
Factor
\[99-45f+36b\] to identify the equivalent expressions.
Choose 2 answers:
Choose 2 answers:
(Choice A)
\[9(11-5f+4b)\]
A
\[9(11-5f+4b)\]
(Choice B)
\[3(33-15+12)\]
B
\[3(33-15+12)\]
(Choice C)
\[11(9-4f+3b)\]
C
\[11(9-4f+3b)\]
(Choice D)
\[3(33-15f+12b)\]
D
\[3(33-15f+12b)\]

Answer 1

To factor the expression \(99 - 45f + 36b\), we first look for the greatest common factor (GCF) of the constants and the coefficients.

1. The coefficients of the terms \(99\), \(-45f\), and \(36b\) are \(99\), \(-45\), and \(36\).
2. The GCF of these numbers can be calculated as follows:
- The factors of \(99\) are \(1, 3, 9, 11, 33, 99\).
- The factors of \(45\) are \(1, 3, 5, 9, 15, 45\).
- The factors of \(36\) are \(1, 2, 3, 4, 6, 9, 12, 18, 36\).
- The common factors among the three numbers are \(1\), \(3\), and \(9\).
- The highest common factor is \(9\).

Next, we can factor out \(9\) from the expression.

Starting with the original expression:
\[
99 - 45f + 36b
\]

We factor out \(9\):
\[
= 9(11 - 5f + 4b)
\]

So after factoring, we have:
\[
99 - 45f + 36b = 9(11 - 5f + 4b)
\]

From the provided choices, we can identify that the equivalent expression obtained by factoring is:

- (Choice A) \(9(11 - 5f + 4b)\)

Next, we verify which other choice can be simplified to match our expression:

(Choice D) is \(3(33 - 15f + 12b)\).
Let's simplify it:
\[
3(33 - 15f + 12b) = 3 \times 33 - 3 \times 15f + 3 \times 12b = 99 - 45f + 36b
\]
This is indeed the original expression we started with before factoring.

Therefore, the other equivalent expression from the choices is:

- (Choice D) \(3(33 - 15f + 12b)\)

Thus, the two correct choices are:
- (Choice A) \(9(11 - 5f + 4b)\)
- (Choice D) \(3(33 - 15f + 12b)\)