Question

factoring expressions
example showing how to use factoring to write an equivalent
form. then solve problems 1–3.
example
consider the expression (-3t + 3) + (2 - 2t) + (-4t + 4). write an
equivalent expression that is the product of two factors.
you can combine like terms. then find a common factor.
(-3t + 3) + (2 - 2t) + (-4t + 4)
(-3t - 2t - 4t) + (3 + 2 + 4)
= -9t + 9
= 9(-t + 1)
an equivalent expression is 9(-t + 1).
1 write -9t + 9 as the product of two factors in a way that is not shown in the
example. explain how you found it.
2 write an expression equivalent to 6 - 4(3 - 6m) + 12m that is the product of two
factors. show your work.
solution
3 is 1 + 4(3x - 10) - 12x equivalent to -39? explain.
vocabulary
equivalent
expressions
two or more different
expressions that
always name the
same value.
factor (noun)
a number, or
expression within
parentheses, that is
multiplied
factor (verb)
to rewrite an
expression as a
product of factors.

Explanation:

Problem 1: Write \(-9t + 9\) as a product of two factors (new way)

Step 1: Find GCF of terms

The terms are \(-9t\) and \(9\). GCF of \(9\) and \(9\) (coefficients) is \(9\), but also consider negative. Let's take \(-9\) or \(9\), or other common factors. Wait, another way: factor out \(-9\) or \(9\), but also, we can factor out \(9(-t + 1)\), but the example had \(9(t - 1)\)? Wait no, original example had \(-9t + 9 = 9(-t + 1)\) or \(-9(t - 1)\). Wait, let's do it differently. Let's factor out \(9\) first: \(-9t + 9 = 9(-t + 1)\). But maybe factor out \(-9\): \(-9(t - 1)\). Wait, the example might have used \(9(-t + 1)\), but let's check. Wait, the expression is \(-9t + 9\). Let's factor out \(9\): \(9(-t + 1)\). Alternatively, factor out \(-9\): \(-9(t - 1)\). Or, factor out \(3\): \(3(-3t + 3)\), but that's not simplest. Wait, the question says "not shown in the example". Let's assume the example used \(9(-t + 1)\), so we can use \(-9(t - 1)\) or \(9(1 - t)\). Let's verify: \(-9(t - 1) = -9t + 9\), correct. So step 1: identify common factor, step 2: factor out.

Step 1: Identify common factors

Terms: \(-9t\) and \(9\). Common factors include \(9\), \(-9\), \(3\), \(-3\), etc. Let's choose \(-9\) as a common factor.

Step 2: Factor out \(-9\)

\(-9t + 9 = -9(t - 1)\) (since \(-9 \times t = -9t\) and \(-9 \times (-1) = 9\)).

Step 1: Distribute \(-4\)

\(6 - 4(3) + 4(6m) + 12m = 6 - 12 + 24m + 12m\)

Step 2: Combine like terms

\(6 - 12 = -6\); \(24m + 12m = 36m\). So expression becomes \(-6 + 36m\)

Step 3: Factor out GCF of \(-6\) and \(36m\), which is \(6\) (or \(-6\))

Factor out \(6\): \(6(-1 + 6m)\) or \(6(6m - 1)\). Let's check: \(6 \times 6m = 36m\), \(6 \times (-1) = -6\), so \(6(6m - 1) = 36m - 6 = -6 + 36m\), correct.

Step 1: Distribute \(4\)

\(1 + 4 \times 3x - 4 \times 10 - 12x = 1 + 12x - 40 - 12x\)

Step 2: Combine like terms

\(12x - 12x = 0\); \(1 - 40 = -39\). So the expression simplifies to \(-39\), so they are equivalent.

Answer:

\(-9(t - 1)\) (or \(9(1 - t)\), but \(-9(t - 1)\) is another valid factorization)

Problem 2: Simplify \(6 - 4(3 - 6m) + 12m\) and factor