Certainly! Let's analyze and match each pair of polynomials with their corresponding sums.
1. We have four pairs of polynomials to match with the given sums. The pairs of polynomials and the given sums are:
Pairs of polynomials:
1. [tex]\( 12x^2 + 3x + 6 \)[/tex] and [tex]\( -7x^2 - 4x - 2 \)[/tex]
2. [tex]\( 2x^2 - x \)[/tex] and [tex]\( -x - 2x^2 - 2 \)[/tex]
3. [tex]\( x + x^2 + 2 \)[/tex] and [tex]\( x^2 - 2 - x \)[/tex]
4. [tex]\( x^2 + x \)[/tex] and [tex]\( x^2 + 8x - 2 \)[/tex]
Given sums:
1. [tex]\(-2x - 2\)[/tex]
2. [tex]\(2x^2\)[/tex]
3. [tex]\(2x^2 + 9x - 2\)[/tex]
4. [tex]\(5x^2 - x + 4\)[/tex]
2. Let's match each pair of polynomials to their corresponding sum:
Pair 1: [tex]\( 12x^2 + 3x + 6 \)[/tex] and [tex]\( -7x^2 - 4x - 2 \)[/tex]
To find the sum:
[tex]\[
12x^2 + 3x + 6 + (-7x^2 - 4x - 2) = (12x^2 - 7x^2) + (3x - 4x) + (6 - 2) = 5x^2 - x + 4
\][/tex]
This matches with the sum: [tex]\( 5x^2 - x + 4 \)[/tex]
Pair 2: [tex]\( 2x^2 - x \)[/tex] and [tex]\( -x - 2x^2 - 2 \)[/tex]
To find the sum:
[tex]\[
2x^2 - x + (-x - 2x^2 - 2) = (2x^2 - 2x^2) + (-x - x) - 2 = -2x - 2
\][/tex]
This matches with the sum: [tex]\( -2x - 2 \)[/tex]
Pair 3: [tex]\( x + x^2 + 2 \)[/tex] and [tex]\( x^2 - 2 - x \)[/tex]
To find the sum:
[tex]\[
x + x^2 + 2 + (x^2 - 2 - x) = (x^2 + x^2) + (x - x) + (2 - 2) = 2x^2
\][/tex]
This matches with the sum: [tex]\( 2x^2 \)[/tex]
Pair 4: [tex]\( x^2 + x \)[/tex] and [tex]\( x^2 + 8x - 2 \)[/tex]
To find the sum:
[tex]\[
x^2 + x + (x^2 + 8x - 2) = (x^2 + x^2) + (x + 8x) - 2 = 2x^2 + 9x - 2
\][/tex]
This matches with the sum: [tex]\( 2x^2 + 9x - 2 \)[/tex]
3. Now, let's summarize the matches:
- Pair [tex]\( (12x^2 + 3x + 6) \)[/tex] and [tex]\( (-7x^2 - 4x - 2) \)[/tex] matches with [tex]\( 5x^2 - x + 4 \)[/tex]
- Pair [tex]\( (2x^2 - x) \)[/tex] and [tex]\( (-x - 2x^2 - 2) \)[/tex] matches with [tex]\( -2x - 2 \)[/tex]
- Pair [tex]\( (x + x^2 + 2) \)[/tex] and [tex]\( (x^2 - 2 - x) \)[/tex] matches with [tex]\( 2x^2 \)[/tex]
- Pair [tex]\( (x^2 + x) \)[/tex] and [tex]\( (x^2 + 8x - 2) \)[/tex] matches with [tex]\( 2x^2 + 9x - 2 \)[/tex]
Thus, the matching of each pair of polynomials to their sum is as follows:
[tex]\[
\begin{align*}
1. & \ (12x^2 + 3x + 6) \ \text{and} \ (-7x^2 - 4x - 2) \to 5x^2 - x + 4 \\
2. & \ (2x^2 - x) \ \text{and} \ (-x - 2x^2 - 2) \to -2x - 2 \\
3. & \ (x + x^2 + 2) \ \text{and} \ (x^2 - 2 - x) \to 2x^2 \\
4. & \ (x^2 + x) \ \text{and} \ (x^2 + 8x - 2) \to 2x^2 + 9x - 2 \\
\end{align*}
\][/tex]
