Answer :
To simplify the given polynomial expression:
[tex]\[
\left(5x^4 - 9x^3 + 7x - 1\right) + \left(-8x^4 + 4x^2 - 3x + 2\right) - \left(-4x^3 + 5x - 1\right)(2x - 7)
\][/tex]
we will go through the following steps:
1. Combine the first two polynomials:
[tex]\[
(5x^4 - 9x^3 + 7x - 1) + (-8x^4 + 4x^2 - 3x + 2)
\][/tex]
Start by combining like terms:
- The [tex]\(x^4\)[/tex] terms: [tex]\(5x^4 - 8x^4 = -3x^4\)[/tex]
- The [tex]\(x^3\)[/tex] terms: [tex]\(-9x^3\)[/tex]
- The [tex]\(x^2\)[/tex] term: [tex]\(+4x^2\)[/tex]
- The [tex]\(x\)[/tex] terms: [tex]\(7x - 3x = 4x\)[/tex]
- The constant terms: [tex]\(-1 + 2 = 1\)[/tex]
This gives us:
[tex]\[
-3x^4 - 9x^3 + 4x^2 + 4x + 1
\][/tex]
2. Expand the product in the third expression:
[tex]\[
\left(-4x^3 + 5x - 1\right)(2x - 7)
\][/tex]
Use distribution (or the distributive property) to expand:
[tex]\[
-4x^3 \cdot 2x = -8x^4
\][/tex]
[tex]\[
-4x^3 \cdot (-7) = 28x^3
\][/tex]
[tex]\[
5x \cdot 2x = 10x^2
\][/tex]
[tex]\[
5x \cdot (-7) = -35x
\][/tex]
[tex]\[
-1 \cdot 2x = -2x
\][/tex]
[tex]\[
-1 \cdot (-7) = 7
\][/tex]
Combine these results:
[tex]\[
-8x^4 + 28x^3 + 10x^2 - 35x - 2x + 7 = -8x^4 + 28x^3 + 10x^2 - 37x + 7
\][/tex]
3. Subtract the expanded polynomial from the combined polynomials:
Subtract:
[tex]\[
(-3x^4 - 9x^3 + 4x^2 + 4x + 1) - (-8x^4 + 28x^3 + 10x^2 - 37x + 7)
\][/tex]
This means we add [tex]\(8x^4\)[/tex], subtract [tex]\(28x^3\)[/tex], subtract [tex]\(10x^2\)[/tex], add [tex]\(37x\)[/tex], and subtract 7 from the first expression:
- The [tex]\(x^4\)[/tex] terms: [tex]\(-3x^4 + 8x^4 = 5x^4\)[/tex]
- The [tex]\(x^3\)[/tex] terms: [tex]\(-9x^3 - 28x^3 = -37x^3\)[/tex]
- The [tex]\(x^2\)[/tex] terms: [tex]\(4x^2 - 10x^2 = -6x^2\)[/tex]
- The [tex]\(x\)[/tex] terms: [tex]\(4x + 37x = 41x\)[/tex]
- The constant terms: [tex]\(1 - 7 = -6\)[/tex]
So, the simplified expression is:
[tex]\[
5x^4 - 37x^3 - 6x^2 + 41x - 6
\][/tex]
The correct answer is:
[tex]\[
\text{A. } 5x^4 - 37x^3 - 6x^2 + 41x - 6
\][/tex]
[tex]\[
\left(5x^4 - 9x^3 + 7x - 1\right) + \left(-8x^4 + 4x^2 - 3x + 2\right) - \left(-4x^3 + 5x - 1\right)(2x - 7)
\][/tex]
we will go through the following steps:
1. Combine the first two polynomials:
[tex]\[
(5x^4 - 9x^3 + 7x - 1) + (-8x^4 + 4x^2 - 3x + 2)
\][/tex]
Start by combining like terms:
- The [tex]\(x^4\)[/tex] terms: [tex]\(5x^4 - 8x^4 = -3x^4\)[/tex]
- The [tex]\(x^3\)[/tex] terms: [tex]\(-9x^3\)[/tex]
- The [tex]\(x^2\)[/tex] term: [tex]\(+4x^2\)[/tex]
- The [tex]\(x\)[/tex] terms: [tex]\(7x - 3x = 4x\)[/tex]
- The constant terms: [tex]\(-1 + 2 = 1\)[/tex]
This gives us:
[tex]\[
-3x^4 - 9x^3 + 4x^2 + 4x + 1
\][/tex]
2. Expand the product in the third expression:
[tex]\[
\left(-4x^3 + 5x - 1\right)(2x - 7)
\][/tex]
Use distribution (or the distributive property) to expand:
[tex]\[
-4x^3 \cdot 2x = -8x^4
\][/tex]
[tex]\[
-4x^3 \cdot (-7) = 28x^3
\][/tex]
[tex]\[
5x \cdot 2x = 10x^2
\][/tex]
[tex]\[
5x \cdot (-7) = -35x
\][/tex]
[tex]\[
-1 \cdot 2x = -2x
\][/tex]
[tex]\[
-1 \cdot (-7) = 7
\][/tex]
Combine these results:
[tex]\[
-8x^4 + 28x^3 + 10x^2 - 35x - 2x + 7 = -8x^4 + 28x^3 + 10x^2 - 37x + 7
\][/tex]
3. Subtract the expanded polynomial from the combined polynomials:
Subtract:
[tex]\[
(-3x^4 - 9x^3 + 4x^2 + 4x + 1) - (-8x^4 + 28x^3 + 10x^2 - 37x + 7)
\][/tex]
This means we add [tex]\(8x^4\)[/tex], subtract [tex]\(28x^3\)[/tex], subtract [tex]\(10x^2\)[/tex], add [tex]\(37x\)[/tex], and subtract 7 from the first expression:
- The [tex]\(x^4\)[/tex] terms: [tex]\(-3x^4 + 8x^4 = 5x^4\)[/tex]
- The [tex]\(x^3\)[/tex] terms: [tex]\(-9x^3 - 28x^3 = -37x^3\)[/tex]
- The [tex]\(x^2\)[/tex] terms: [tex]\(4x^2 - 10x^2 = -6x^2\)[/tex]
- The [tex]\(x\)[/tex] terms: [tex]\(4x + 37x = 41x\)[/tex]
- The constant terms: [tex]\(1 - 7 = -6\)[/tex]
So, the simplified expression is:
[tex]\[
5x^4 - 37x^3 - 6x^2 + 41x - 6
\][/tex]
The correct answer is:
[tex]\[
\text{A. } 5x^4 - 37x^3 - 6x^2 + 41x - 6
\][/tex]
