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Answer :
To simplify the given polynomial expression [tex]\((5x^4 - 9x^3 + 7x - 1) + (-8x^4 + 4x^2 - 3x + 2) - (-4x^3 + 5x - 1)(2x - 7)\)[/tex], we need to perform the following steps:
1. Combine the first two polynomials:
- The first polynomial is [tex]\(5x^4 - 9x^3 + 7x - 1\)[/tex].
- The second polynomial is [tex]\(-8x^4 + 4x^2 - 3x + 2\)[/tex].
Adding these two expressions together:
[tex]\[
(5x^4 - 9x^3 + 7x - 1) + (-8x^4 + 4x^2 - 3x + 2)
\][/tex]
Combine like terms:
- [tex]\(5x^4 - 8x^4 = -3x^4\)[/tex]
- [tex]\(-9x^3\)[/tex]
- [tex]\(+ 4x^2\)[/tex]
- [tex]\(7x - 3x = 4x\)[/tex]
- [tex]\(-1 + 2 = 1\)[/tex]
This gives us:
[tex]\(-3x^4 - 9x^3 + 4x^2 + 4x + 1\)[/tex].
2. Expand and simplify the product in the third part:
- [tex]\((-4x^3 + 5x - 1)(2x - 7)\)[/tex].
Distribute each term:
- [tex]\(-4x^3 \times 2x = -8x^4\)[/tex]
- [tex]\(-4x^3 \times -7 = 28x^3\)[/tex]
- [tex]\(5x \times 2x = 10x^2\)[/tex]
- [tex]\(5x \times -7 = -35x\)[/tex]
- [tex]\(-1 \times 2x = -2x\)[/tex]
- [tex]\(-1 \times -7 = 7\)[/tex]
Adding these terms gives:
[tex]\(-8x^4 + 28x^3 + 10x^2 - 35x - 2x + 7\)[/tex].
Combine like terms:
- [tex]\(-8x^4\)[/tex]
- [tex]\(28x^3\)[/tex]
- [tex]\(10x^2\)[/tex]
- [tex]\(-35x - 2x = -37x\)[/tex]
- [tex]\(+ 7\)[/tex]
So it simplifies to:
[tex]\(-8x^4 + 28x^3 + 10x^2 - 37x + 7\)[/tex].
3. Subtract the expanded product from the result of step 1:
We have [tex]\(-3x^4 - 9x^3 + 4x^2 + 4x + 1\)[/tex], and we need to subtract:
[tex]\(-8x^4 + 28x^3 + 10x^2 - 37x + 7\)[/tex].
Perform the subtraction:
- [tex]\((-3x^4) - (-8x^4) = 5x^4\)[/tex]
- [tex]\((-9x^3) - 28x^3 = -37x^3\)[/tex]
- [tex]\(4x^2 - 10x^2 = -6x^2\)[/tex]
- [tex]\(4x - (-37x) = 41x\)[/tex]
- [tex]\(1 - 7 = -6\)[/tex]
The final simplified expression is:
[tex]\[
5x^4 - 37x^3 - 6x^2 + 41x - 6
\][/tex]
So, the correct answer is [tex]\(\boxed{5x^4 - 37x^3 - 6x^2 + 41x - 6}\)[/tex], which corresponds to option B.
1. Combine the first two polynomials:
- The first polynomial is [tex]\(5x^4 - 9x^3 + 7x - 1\)[/tex].
- The second polynomial is [tex]\(-8x^4 + 4x^2 - 3x + 2\)[/tex].
Adding these two expressions together:
[tex]\[
(5x^4 - 9x^3 + 7x - 1) + (-8x^4 + 4x^2 - 3x + 2)
\][/tex]
Combine like terms:
- [tex]\(5x^4 - 8x^4 = -3x^4\)[/tex]
- [tex]\(-9x^3\)[/tex]
- [tex]\(+ 4x^2\)[/tex]
- [tex]\(7x - 3x = 4x\)[/tex]
- [tex]\(-1 + 2 = 1\)[/tex]
This gives us:
[tex]\(-3x^4 - 9x^3 + 4x^2 + 4x + 1\)[/tex].
2. Expand and simplify the product in the third part:
- [tex]\((-4x^3 + 5x - 1)(2x - 7)\)[/tex].
Distribute each term:
- [tex]\(-4x^3 \times 2x = -8x^4\)[/tex]
- [tex]\(-4x^3 \times -7 = 28x^3\)[/tex]
- [tex]\(5x \times 2x = 10x^2\)[/tex]
- [tex]\(5x \times -7 = -35x\)[/tex]
- [tex]\(-1 \times 2x = -2x\)[/tex]
- [tex]\(-1 \times -7 = 7\)[/tex]
Adding these terms gives:
[tex]\(-8x^4 + 28x^3 + 10x^2 - 35x - 2x + 7\)[/tex].
Combine like terms:
- [tex]\(-8x^4\)[/tex]
- [tex]\(28x^3\)[/tex]
- [tex]\(10x^2\)[/tex]
- [tex]\(-35x - 2x = -37x\)[/tex]
- [tex]\(+ 7\)[/tex]
So it simplifies to:
[tex]\(-8x^4 + 28x^3 + 10x^2 - 37x + 7\)[/tex].
3. Subtract the expanded product from the result of step 1:
We have [tex]\(-3x^4 - 9x^3 + 4x^2 + 4x + 1\)[/tex], and we need to subtract:
[tex]\(-8x^4 + 28x^3 + 10x^2 - 37x + 7\)[/tex].
Perform the subtraction:
- [tex]\((-3x^4) - (-8x^4) = 5x^4\)[/tex]
- [tex]\((-9x^3) - 28x^3 = -37x^3\)[/tex]
- [tex]\(4x^2 - 10x^2 = -6x^2\)[/tex]
- [tex]\(4x - (-37x) = 41x\)[/tex]
- [tex]\(1 - 7 = -6\)[/tex]
The final simplified expression is:
[tex]\[
5x^4 - 37x^3 - 6x^2 + 41x - 6
\][/tex]
So, the correct answer is [tex]\(\boxed{5x^4 - 37x^3 - 6x^2 + 41x - 6}\)[/tex], which corresponds to option B.
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