Question

factor \\(99 - 45f + 36b\\) to identify the equivalent expressions.

choose 2 answers:

a \\(9(11 - 5f + 4b)\\)

b \\(3(33 - 15 + 12)\\)

c \\(11(9 - 4f + 3b)\\)

d \\(3(33 - 15f + 12b)\\)

Explanation:

Find common factors of the terms

To factor the expression \(99 - 45f + 36b\), we first find the common factors of the numerical coefficients: \(99\), \(45\), and \(36\).
Using the Greatest Common Factor concept, we list the factors:

• Factors of \(99\): \(1, 3, 9, 11, 33, 99\)
• Factors of \(45\): \(1, 3, 5, 9, 15, 45\)
• Factors of \(36\): \(1, 2, 3, 4, 6, 9, 12, 18, 36\)

The common factors are \(1\), \(3\), and \(9\).

Factor out the greatest common factor

Using the Distributive Property in reverse, we can factor out the greatest common factor, which is \(9\):

$$ 99 - 45f + 36b = 9(11) - 9(5f) + 9(4b) $$

Factoring out \(9\) gives:

$$ 9(11 - 5f + 4b) $$

This matches option (A).

Factor out other common factors

We can also factor out another common factor, \(3\), to find another Equivalent Expression:

$$ 99 - 45f + 36b = 3(33) - 3(15f) + 3(12b) $$

Factoring out \(3\) gives:

$$ 3(33 - 15f + 12b) $$

This matches option (D).

Verify the remaining options

Let's expand the remaining choices to ensure they are not equivalent:

• For (B): \(3(33 - 15 + 12) = 3(30) = 90\), which is missing the variables \(f\) and \(b\).
• For (C): \(11(9 - 4f + 3b) = 99 - 44f + 33b\), which does not match our original expression.

Answer:

• (A) \(9(11 - 5f + 4b)\) (Correct answer)
• (B) \(3(33 - 15 + 12)\)
• (C) \(11(9 - 4f + 3b)\)
• (D) \(3(33 - 15f + 12b)\) (Correct answer)