Answer :
To simplify the given polynomial expression, we need to perform the operations correctly and combine like terms. Here's how you can do it step by step:
1. Identify Each Polynomial:
- The first polynomial is: [tex]\(5x^4 - 9x^3 + 7x - 1\)[/tex]
- The second polynomial is: [tex]\(-8x^4 + 4x^2 - 3x + 2\)[/tex]
- The expression to subtract is: [tex]\((-4x^3 + 5x - 1)(2x - 7)\)[/tex]
2. Expand the Expression to Subtract:
First, let's expand [tex]\((-4x^3 + 5x - 1)(2x - 7)\)[/tex].
[tex]\[
(-4x^3 + 5x - 1) \times (2x - 7) = -4x^3 \times 2x + 5x \times 2x - 1 \times 2x -4x^3 \times (-7) + 5x \times (-7) - 1 \times (-7)
\][/tex]
[tex]\[
= -8x^4 + 10x^2 - 2x + 28x^3 - 35x + 7
\][/tex]
[tex]\[
= -8x^4 + 28x^3 + 10x^2 - 37x + 7
\][/tex]
3. Substitute and Simplify All Expressions:
Substitute this expanded result into the original expression:
[tex]\[
\left(5x^4 - 9x^3 + 7x - 1\right) + \left(-8x^4 + 4x^2 - 3x + 2\right) - \left(-8x^4 + 28x^3 + 10x^2 - 37x + 7\right)
\][/tex]
4. Combine Like Terms:
- Combine [tex]\(x^4\)[/tex] terms: [tex]\(5x^4 + (-8x^4) - (-8x^4) = 5x^4\)[/tex]
- Combine [tex]\(x^3\)[/tex] terms: [tex]\(-9x^3 + 28x^3 = 19x^3\)[/tex]
- Combine [tex]\(x^2\)[/tex] terms: [tex]\(4x^2 - 10x^2 = -6x^2\)[/tex]
- Combine [tex]\(x\)[/tex] terms: [tex]\(7x - 3x + 37x = 41x\)[/tex]
- Combine constant terms: [tex]\(-1 + 2 - 7 = -6\)[/tex]
So, the simplified expression will be:
[tex]\[
5x^4 - 37x^3 - 6x^2 + 41x - 6
\][/tex]
The correct answer, matched to the options provided, is:
C. [tex]\(5x^4 - 37x^3 - 6x^2 + 41x - 6\)[/tex]
1. Identify Each Polynomial:
- The first polynomial is: [tex]\(5x^4 - 9x^3 + 7x - 1\)[/tex]
- The second polynomial is: [tex]\(-8x^4 + 4x^2 - 3x + 2\)[/tex]
- The expression to subtract is: [tex]\((-4x^3 + 5x - 1)(2x - 7)\)[/tex]
2. Expand the Expression to Subtract:
First, let's expand [tex]\((-4x^3 + 5x - 1)(2x - 7)\)[/tex].
[tex]\[
(-4x^3 + 5x - 1) \times (2x - 7) = -4x^3 \times 2x + 5x \times 2x - 1 \times 2x -4x^3 \times (-7) + 5x \times (-7) - 1 \times (-7)
\][/tex]
[tex]\[
= -8x^4 + 10x^2 - 2x + 28x^3 - 35x + 7
\][/tex]
[tex]\[
= -8x^4 + 28x^3 + 10x^2 - 37x + 7
\][/tex]
3. Substitute and Simplify All Expressions:
Substitute this expanded result into the original expression:
[tex]\[
\left(5x^4 - 9x^3 + 7x - 1\right) + \left(-8x^4 + 4x^2 - 3x + 2\right) - \left(-8x^4 + 28x^3 + 10x^2 - 37x + 7\right)
\][/tex]
4. Combine Like Terms:
- Combine [tex]\(x^4\)[/tex] terms: [tex]\(5x^4 + (-8x^4) - (-8x^4) = 5x^4\)[/tex]
- Combine [tex]\(x^3\)[/tex] terms: [tex]\(-9x^3 + 28x^3 = 19x^3\)[/tex]
- Combine [tex]\(x^2\)[/tex] terms: [tex]\(4x^2 - 10x^2 = -6x^2\)[/tex]
- Combine [tex]\(x\)[/tex] terms: [tex]\(7x - 3x + 37x = 41x\)[/tex]
- Combine constant terms: [tex]\(-1 + 2 - 7 = -6\)[/tex]
So, the simplified expression will be:
[tex]\[
5x^4 - 37x^3 - 6x^2 + 41x - 6
\][/tex]
The correct answer, matched to the options provided, is:
C. [tex]\(5x^4 - 37x^3 - 6x^2 + 41x - 6\)[/tex]
