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2. Type the correct answer in each box.

Consider the expressions shown below.

\begin{tabular}{|c|c|c|}
\hline A & B & C \\
\hline [tex]$-7x^2-2x+5$[/tex] & [tex]$7x^2-2x+7$[/tex] & [tex]$7x^2+2x-5$[/tex] \\
\hline
\end{tabular}

Complete each of the following statements with the letter that represents the expression.

[tex]\[
(3x^2-6x+11)-(10x^2-4x+6) \text{ is equivalent to expression } \square
\][/tex]

[tex]\[
(-3x^2-5x-3)-(-10x^2-7x+2) \text{ is equivalent to expression } \square
\][/tex]

[tex]\[
(12x^2+6x-5)-(5x^2+8x-12) \text{ is equivalent to expression } \square
\][/tex]



Answer :

GinnyAnswer
Let's solve each of the given expressions step by step:

1. Evaluate [tex]\((3x^2 - 6x + 11) - (10x^2 - 4x + 6)\)[/tex]:
- Distribute the negative sign through the second polynomial: [tex]\(3x^2 - 6x + 11 - 10x^2 + 4x - 6\)[/tex]
- Combine like terms:
- [tex]\(3x^2 - 10x^2 = -7x^2\)[/tex]
- [tex]\(-6x + 4x = -2x\)[/tex]
- [tex]\(11 - 6 = 5\)[/tex]
- The resulting expression is [tex]\(-7x^2 - 2x + 5\)[/tex]
- This matches expression [tex]\(A\)[/tex].

2. Evaluate [tex]\((-3x^2 - 5x - 3) - (-10x^2 - 7x + 2)\)[/tex]:
- Distribute the negative sign through the second polynomial: [tex]\(-3x^2 - 5x - 3 + 10x^2 + 7x - 2\)[/tex]
- Combine like terms:
- [tex]\(-3x^2 + 10x^2 = 7x^2\)[/tex]
- [tex]\(-5x + 7x = 2x\)[/tex]
- [tex]\(-3 - 2 = -5\)[/tex]
- The resulting expression is [tex]\(7x^2 + 2x - 5\)[/tex]
- This matches expression [tex]\(C\)[/tex].

3. Evaluate [tex]\((12x^2 + 6x - 5) - (5x^2 + 8x - 12)\)[/tex]:
- Distribute the negative sign through the second polynomial: [tex]\(12x^2 + 6x - 5 - 5x^2 - 8x + 12\)[/tex]
- Combine like terms:
- [tex]\(12x^2 - 5x^2 = 7x^2\)[/tex]
- [tex]\(6x - 8x = -2x\)[/tex]
- [tex]\(-5 + 12 = 7\)[/tex]
- The resulting expression is [tex]\(7x^2 - 2x + 7\)[/tex]
- This matches expression [tex]\(B\)[/tex].

So the completed statements are:

1. [tex]\((3x^2 - 6x + 11) - (10x^2 - 4x + 6)\)[/tex] is equivalent to expression [tex]\(\boxed{A}\)[/tex]
2. [tex]\((-3x^2 - 5x - 3) - (-10x^2 - 7x + 2)\)[/tex] is equivalent to expression [tex]\(\boxed{C}\)[/tex]
3. [tex]\((12x^2 + 6x - 5) - (5x^2 + 8x - 12)\)[/tex] is equivalent to expression [tex]\(\boxed{B}\)[/tex]

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