Answer :
To simplify the given polynomial expression step-by-step, we'll follow these steps:
We're given the expression:
[tex]\[
(5x^4 - 9x^3 + 7x - 1) + (-8x^4 + 4x^2 - 3x + 2) - (-4x^3 + 5x - 1)(2x - 7)
\][/tex]
1. Combine the first two polynomials:
[tex]\[
(5x^4 - 9x^3 + 7x - 1) + (-8x^4 + 4x^2 - 3x + 2)
\][/tex]
Combine like terms:
- [tex]\( x^4 \)[/tex] terms: [tex]\( 5x^4 - 8x^4 = -3x^4 \)[/tex]
- [tex]\( x^3 \)[/tex] terms: [tex]\( -9x^3 \)[/tex]
- [tex]\( x^2 \)[/tex] terms: [tex]\( +4x^2 \)[/tex]
- [tex]\( x \)[/tex] terms: [tex]\( 7x - 3x = 4x \)[/tex]
- Constant terms: [tex]\( -1 + 2 = 1 \)[/tex]
So, this part simplifies to:
[tex]\[
-3x^4 - 9x^3 + 4x^2 + 4x + 1
\][/tex]
2. Simplify the product and subtraction of the third expression:
- We're given [tex]\((-4x^3 + 5x - 1)(2x - 7)\)[/tex].
Distribute the terms:
- [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 to get:
[tex]\(-8x^4 + 28x^3 + 10x^2 - 35x - 2x + 7 = -8x^4 + 28x^3 + 10x^2 - 37x + 7\)[/tex]
3. Subtract this expression from the first result:
[tex]\((-3x^4 - 9x^3 + 4x^2 + 4x + 1) - (-8x^4 + 28x^3 + 10x^2 - 37x + 7)\)[/tex]
Distribute the negative sign and combine like terms:
- [tex]\( x^4 \)[/tex] terms: [tex]\(-3x^4 - (-8x^4) = -3x^4 + 8x^4 = 5x^4\)[/tex]
- [tex]\( x^3 \)[/tex] terms: [tex]\(-9x^3 - 28x^3 = -37x^3\)[/tex]
- [tex]\( x^2 \)[/tex] terms: [tex]\(4x^2 - 10x^2 = -6x^2\)[/tex]
- [tex]\( x \)[/tex] terms: [tex]\(4x - (-37x) = 4x + 37x = 41x\)[/tex]
- Constant terms: [tex]\(1 - 7 = -6\)[/tex]
Combining everything, the simplified polynomial expression is:
[tex]\[
5x^4 - 37x^3 - 6x^2 + 41x - 6
\][/tex]
Thus, the correct answer is option:
A. [tex]\(5x^4 - 37x^3 - 6x^2 + 41x - 6\)[/tex]
We're given the expression:
[tex]\[
(5x^4 - 9x^3 + 7x - 1) + (-8x^4 + 4x^2 - 3x + 2) - (-4x^3 + 5x - 1)(2x - 7)
\][/tex]
1. Combine the first two polynomials:
[tex]\[
(5x^4 - 9x^3 + 7x - 1) + (-8x^4 + 4x^2 - 3x + 2)
\][/tex]
Combine like terms:
- [tex]\( x^4 \)[/tex] terms: [tex]\( 5x^4 - 8x^4 = -3x^4 \)[/tex]
- [tex]\( x^3 \)[/tex] terms: [tex]\( -9x^3 \)[/tex]
- [tex]\( x^2 \)[/tex] terms: [tex]\( +4x^2 \)[/tex]
- [tex]\( x \)[/tex] terms: [tex]\( 7x - 3x = 4x \)[/tex]
- Constant terms: [tex]\( -1 + 2 = 1 \)[/tex]
So, this part simplifies to:
[tex]\[
-3x^4 - 9x^3 + 4x^2 + 4x + 1
\][/tex]
2. Simplify the product and subtraction of the third expression:
- We're given [tex]\((-4x^3 + 5x - 1)(2x - 7)\)[/tex].
Distribute the terms:
- [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 to get:
[tex]\(-8x^4 + 28x^3 + 10x^2 - 35x - 2x + 7 = -8x^4 + 28x^3 + 10x^2 - 37x + 7\)[/tex]
3. Subtract this expression from the first result:
[tex]\((-3x^4 - 9x^3 + 4x^2 + 4x + 1) - (-8x^4 + 28x^3 + 10x^2 - 37x + 7)\)[/tex]
Distribute the negative sign and combine like terms:
- [tex]\( x^4 \)[/tex] terms: [tex]\(-3x^4 - (-8x^4) = -3x^4 + 8x^4 = 5x^4\)[/tex]
- [tex]\( x^3 \)[/tex] terms: [tex]\(-9x^3 - 28x^3 = -37x^3\)[/tex]
- [tex]\( x^2 \)[/tex] terms: [tex]\(4x^2 - 10x^2 = -6x^2\)[/tex]
- [tex]\( x \)[/tex] terms: [tex]\(4x - (-37x) = 4x + 37x = 41x\)[/tex]
- Constant terms: [tex]\(1 - 7 = -6\)[/tex]
Combining everything, the simplified polynomial expression is:
[tex]\[
5x^4 - 37x^3 - 6x^2 + 41x - 6
\][/tex]
Thus, the correct answer is option:
A. [tex]\(5x^4 - 37x^3 - 6x^2 + 41x - 6\)[/tex]
