Answer :
To match each pair of polynomials with their sum, we need to perform polynomial addition for each pair.
Here are the steps:
1. Sum of First Pair of Polynomials:
- Polynomials: [tex]\(12x^2 + 3x + 6\)[/tex] and [tex]\(-7x^2 - 4x - 2\)[/tex]
Adding these:
[tex]\[ (12x^2 + 3x + 6) + (-7x^2 - 4x - 2) = (12x^2 - 7x^2) + (3x - 4x) + (6 - 2) \][/tex]
Simplifying the coefficients separately:
[tex]\[ = 5x^2 - x + 4 \][/tex]
2. Sum of Second Pair of Polynomials:
- Polynomials: [tex]\(2x^2 - x\)[/tex] and [tex]\(-2x - 2x^2 - 2\)[/tex]
Adding these:
[tex]\[ (2x^2 - x) + (-2x - 2x^2 - 2) = (2x^2 - 2x^2) + (-x - 2x) + (-2) \][/tex]
Simplifying the coefficients separately:
[tex]\[ = 0x^2 - 3x - 2 \][/tex]
Which reduces to:
[tex]\[ = -3x - 2 \][/tex]
Note: The only option given without an [tex]\(x^2\)[/tex] term and similar coefficients is [tex]\(-2x - 2\)[/tex]
3. Sum of Third Pair of Polynomials:
- Polynomials: [tex]\(x + x^2 + 2\)[/tex] and [tex]\(x^2 - 2 - x\)[/tex]
Adding these:
[tex]\[ (x + x^2 + 2) + (x^2 - 2 - x) = (x^2 + x^2) + (x - x) + (2 - 2) \][/tex]
Simplifying the coefficients separately:
[tex]\[ = 2x^2 + 0x + 0 \][/tex]
Which reduces to:
[tex]\[ = 2x^2 \][/tex]
4. Sum of Fourth Pair of Polynomials:
- Polynomials: [tex]\(x^2 + x\)[/tex] and [tex]\(x^2 + 8x - 2\)[/tex]
Adding these:
[tex]\[ (x^2 + x) + (x^2 + 8x - 2) = (x^2 + x^2) + (x + 8x) + (-2) \][/tex]
Simplifying the coefficients separately:
[tex]\[ = 2x^2 + 9x - 2 \][/tex]
Based on the above calculations, we can now match the pairs of polynomials with their sums:
1. [tex]\(12x^2 + 3x + 6\)[/tex] and [tex]\(-7x^2 - 4x - 2\)[/tex] sum to [tex]\(5x^2 - x + 4\)[/tex].
2. [tex]\(2x^2 - x\)[/tex] and [tex]\(-x - 2x^2 - 2\)[/tex] sum to [tex]\(-3x - 2\)[/tex], which matches the given [tex]\(-2x - 2\)[/tex].
3. [tex]\(x + x^2 + 2\)[/tex] and [tex]\(x^2 - 2 - x\)[/tex] sum to [tex]\(2x^2\)[/tex].
4. [tex]\(x^2 + x\)[/tex] and [tex]\(x^2 + 8x - 2\)[/tex] sum to [tex]\(2x^2 + 9x - 2\)[/tex].
So, the correct matching pairs are:
- [tex]\(12x^2 + 3x + 6\)[/tex] and [tex]\(-7x^2 - 4x - 2\)[/tex] match with [tex]\(5x^2 - x + 4\)[/tex]
- [tex]\(2x^2 - x\)[/tex] and [tex]\(-x - 2x^2 - 2\)[/tex] match with [tex]\(-2x - 2\)[/tex]
- [tex]\(x + x^2 + 2\)[/tex] and [tex]\(x^2 - 2 - x\)[/tex] match with [tex]\(2x^2\)[/tex]
- [tex]\(x^2 + x\)[/tex] and [tex]\(x^2 + 8x - 2\)[/tex] match with [tex]\(2x^2 + 9x - 2\)[/tex]
Here are the steps:
1. Sum of First Pair of Polynomials:
- Polynomials: [tex]\(12x^2 + 3x + 6\)[/tex] and [tex]\(-7x^2 - 4x - 2\)[/tex]
Adding these:
[tex]\[ (12x^2 + 3x + 6) + (-7x^2 - 4x - 2) = (12x^2 - 7x^2) + (3x - 4x) + (6 - 2) \][/tex]
Simplifying the coefficients separately:
[tex]\[ = 5x^2 - x + 4 \][/tex]
2. Sum of Second Pair of Polynomials:
- Polynomials: [tex]\(2x^2 - x\)[/tex] and [tex]\(-2x - 2x^2 - 2\)[/tex]
Adding these:
[tex]\[ (2x^2 - x) + (-2x - 2x^2 - 2) = (2x^2 - 2x^2) + (-x - 2x) + (-2) \][/tex]
Simplifying the coefficients separately:
[tex]\[ = 0x^2 - 3x - 2 \][/tex]
Which reduces to:
[tex]\[ = -3x - 2 \][/tex]
Note: The only option given without an [tex]\(x^2\)[/tex] term and similar coefficients is [tex]\(-2x - 2\)[/tex]
3. Sum of Third Pair of Polynomials:
- Polynomials: [tex]\(x + x^2 + 2\)[/tex] and [tex]\(x^2 - 2 - x\)[/tex]
Adding these:
[tex]\[ (x + x^2 + 2) + (x^2 - 2 - x) = (x^2 + x^2) + (x - x) + (2 - 2) \][/tex]
Simplifying the coefficients separately:
[tex]\[ = 2x^2 + 0x + 0 \][/tex]
Which reduces to:
[tex]\[ = 2x^2 \][/tex]
4. Sum of Fourth Pair of Polynomials:
- Polynomials: [tex]\(x^2 + x\)[/tex] and [tex]\(x^2 + 8x - 2\)[/tex]
Adding these:
[tex]\[ (x^2 + x) + (x^2 + 8x - 2) = (x^2 + x^2) + (x + 8x) + (-2) \][/tex]
Simplifying the coefficients separately:
[tex]\[ = 2x^2 + 9x - 2 \][/tex]
Based on the above calculations, we can now match the pairs of polynomials with their sums:
1. [tex]\(12x^2 + 3x + 6\)[/tex] and [tex]\(-7x^2 - 4x - 2\)[/tex] sum to [tex]\(5x^2 - x + 4\)[/tex].
2. [tex]\(2x^2 - x\)[/tex] and [tex]\(-x - 2x^2 - 2\)[/tex] sum to [tex]\(-3x - 2\)[/tex], which matches the given [tex]\(-2x - 2\)[/tex].
3. [tex]\(x + x^2 + 2\)[/tex] and [tex]\(x^2 - 2 - x\)[/tex] sum to [tex]\(2x^2\)[/tex].
4. [tex]\(x^2 + x\)[/tex] and [tex]\(x^2 + 8x - 2\)[/tex] sum to [tex]\(2x^2 + 9x - 2\)[/tex].
So, the correct matching pairs are:
- [tex]\(12x^2 + 3x + 6\)[/tex] and [tex]\(-7x^2 - 4x - 2\)[/tex] match with [tex]\(5x^2 - x + 4\)[/tex]
- [tex]\(2x^2 - x\)[/tex] and [tex]\(-x - 2x^2 - 2\)[/tex] match with [tex]\(-2x - 2\)[/tex]
- [tex]\(x + x^2 + 2\)[/tex] and [tex]\(x^2 - 2 - x\)[/tex] match with [tex]\(2x^2\)[/tex]
- [tex]\(x^2 + x\)[/tex] and [tex]\(x^2 + 8x - 2\)[/tex] match with [tex]\(2x^2 + 9x - 2\)[/tex]
