Let's solve each problem step-by-step and match the results accordingly:
1. Addition of Polynomials:
[tex]\[
(2x^2 + 3x - 1) + (x^2 - 2x + 3)
\][/tex]
Combine like terms:
[tex]\[
2x^2 + x^2 + 3x - 2x - 1 + 3
\][/tex]
Simplify:
[tex]\[
3x^2 + x + 2
\][/tex]
So, the result of the addition is:
[tex]\[
3x^2 + x + 2
\][/tex]
2. Finding the Opposite of a Polynomial:
[tex]\[
3x^2 - x - 4
\][/tex]
Change the signs of all terms:
[tex]\[
-3x^2 + x + 4
\][/tex]
So, the opposite of the polynomial is:
[tex]\[
-3x^2 + x + 4
\][/tex]
3. Subtraction of Polynomials:
[tex]\[
(x^2 - 2x + 3) - (4x^2 + 3x - 1)
\][/tex]
Distribute the negative sign and combine like terms:
[tex]\[
x^2 - 2x + 3 - 4x^2 - 3x + 1
\][/tex]
Simplify:
[tex]\[
x^2 - 4x^2 - 2x - 3x + 3 + 1
\][/tex]
[tex]\[
-3x^2 - 5x + 4
\][/tex]
So, the result of the subtraction is:
[tex]\[
-3x^2 - 5x + 4
\][/tex]
Given these results, we can match them with the options provided:
- Addition:
[tex]\[
(2x^2 + 3x - 1) + (x^2 - 2x + 3) = 3x^2 + x + 2
\][/tex]
So, match with [tex]\(3x^2 + x + 2\)[/tex].
- Opposite:
[tex]\[
\text{The opposite of } 3x^2 - x - 4 \text{ is } -3x^2 + x + 4
\][/tex]
So, match with [tex]\(-3x^2 + x + 4\)[/tex].
- Subtraction:
[tex]\[
(x^2 - 2x + 3) - (4x^2 + 3x - 1) = -3x^2 - 5x + 4
\][/tex]
So, match with [tex]\(-3x^2 - 5x + 4\)[/tex].
Summary of Matching:
1. [tex]\((2x^2 + 3x - 1) + (x^2 - 2x + 3) = 3x^2 + x + 2\)[/tex]
[tex]\[
\text{matches with } 3x^2 + x + 2
\][/tex]
2. [tex]\(-3x^2 + x + 4 \text{ matches with } -3x^2 + x + 4\)[/tex]
3. [tex]\((x^2 - 2x + 3) - (4x^2 + 3x - 1) = -3x^2 - 5x + 4\)[/tex]
[tex]\[
\text{matches with } -3x^2 - 5x + 4
\][/tex]
