Question

write -9t + 9 as the product of two factors in a way that is not shown in the example. explain how you found it.

Explanation:

Assuming the expression is \(-9t + 9\) (since the image text seems to have a typo and likely means this linear expression or maybe a quadratic like \(t^2 - 9t + 9\), but let's assume the linear one first for clarity, or if it's quadratic, we can adjust). Wait, the original text in the image: "Write -9t + 9 as the product of two factors in a way that is not shown in the Example. Explain how you found it."

So let's proceed with \(-9t + 9\).

Step1: Identify the GCF

The greatest common factor (GCF) of \(-9t\) and \(9\) is \(9\), but we can also factor out \(-9\) or other common factors. Wait, let's factor out \(-9\) or \(9\) or a different common factor. Wait, \(-9t + 9 = 9(-t + 1)\) or \(-9(t - 1)\) or also, we can factor out \(3\): \(3(-3t + 3)\), but maybe the example had a different factoring. Let's do it step by step.

Step1: Analyze the terms

The expression is \(-9t + 9\). Let's factor out a common factor. Let's take \(9\) as a common factor: \(9(-t + 1)\). Alternatively, factor out \(-9\): \(-9(t - 1)\). Or factor out \(3\): \(3(-3t + 3)\), but maybe the example had factoring out \(9\) as \(9(1 - t)\), so to do a different one, let's factor out \(-3\): \(-3(3t - 3)\). Wait, no, let's check.

Wait, let's do it properly. The expression is \(-9t + 9\). Let's factor out \(-9\):

Step1: Factor out \(-9\)

Take \(-9\) as the common factor. Then \(-9t + 9 = -9(t) + (-9)(-1) = -9(t - 1)\).

Step2: Verify

Multiply \(-9\) and \((t - 1)\): \(-9 \times t + (-9) \times (-1) = -9t + 9\), which matches the original expression.

Alternatively, factor out \(9\): \(9(-t + 1)\), which is also correct.

Let's confirm. If the example had factored out \(9\) as \(9(1 - t)\) (since \( -t + 1 = 1 - t\)), then to do a different factoring, we can factor out \(-3\):

Step1: Factor out \(-3\)

\(-9t + 9 = -3(3t) + (-3)(-3) = -3(3t - 3)\). Wait, but \(3t - 3\) can be further factored, but the question just asks for two factors, so this is valid.

But let's use the first method: factoring out \(-9\) to get \(-9(t - 1)\).

Answer:

The expression \(-9t + 9\) can be factored as \(\boldsymbol{-9(t - 1)}\). To find this, we identified that both terms \(-9t\) and \(9\) have a common factor of \(-9\). Factoring out \(-9\) from each term: \(-9t = -9 \times t\) and \(9 = -9 \times (-1)\). Using the distributive property (reverse of \(a(b + c) = ab + ac\)), we get \(-9(t - 1)\). Verifying by multiplying \(-9\) with \((t - 1)\) gives \(-9t + 9\), which matches the original expression.