Chapter 7: Problem 23

Show that \(3-2 t-t^{2}=-(t+3)(t-1)\)

Short Answer

Expert verified

Question: Show that the given expression \(3-2t-t^{2}\) can be factored as \(-(t+3)(t-1)\). Answer: After expanding \(-(t+3)(t-1)\), we obtain the equivalent expression \(-(t^2 + 2t - 3)\), which is equal to the original expression \(3-2t-t^2\). This confirms that \(3-2t-t^{2} = -(t+3)(t-1)\).

Step by step solution

Expand the right side expression

We first expand \(-(t+3)(t-1)\) by using the distributive property (also known as FOIL).

Apply distributive property to each term

To use the distributive property, we multiply each term in the first parentheses by each term in the second. Here we get: \((t+3)(t-1) = t(t-1) + 3(t-1)\)

Simplify the expression obtained in step 2

Next, we simplify the expressions found in step 2, using the distributive property again: \(t(t-1) = t^2 - t\) \(3(t-1) = 3t - 3\) Now add the two simplified expressions: \((t^2 - t) + (3t - 3) = t^2 - t + 3t - 3\) Combine like terms: \(t^2 - t + 3t - 3 = t^2 + 2t - 3\)

Add the negative sign

Now, we have to remember to add the negative sign back into the expression. So, we have: \(-(t^2 + 2t - 3)\)

Compare the expanded expression to the original left-hand side expression

Finally, we compare the expanded expression to the original left-hand side expression: Original expression: \(3 - 2t - t^{2}\) Expanded expression: \(-(t^2 + 2t - 3)\) Since we can see that the two expressions are equivalent, we have successfully shown that \(3-2 t-t^{2}=-(t+3)(t-1)\).

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Show that \(3-2 t-t^{2}=-(t+3)(t-1)\)

Short Answer

Step by step solution

Expand the right side expression

Apply distributive property to each term

Simplify the expression obtained in step 2

Add the negative sign

Compare the expanded expression to the original left-hand side expression

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