Answer :
To simplify the given polynomial expression, let's break it down step by step:
We have the expression:
[tex]\[
(5x^4 - 9x^3 + 7x - 1) + (-8x^4 + 4x^2 - 3x + 2) - \left((-4x^3 + 5x - 1)(2x - 7)\right)
\][/tex]
1. Combine like terms from the first two polynomials:
[tex]\[
(5x^4 - 9x^3 + 7x - 1) + (-8x^4 + 4x^2 - 3x + 2)
\][/tex]
Combine these:
- For [tex]\(x^4\)[/tex] terms: [tex]\(5x^4 - 8x^4 = -3x^4\)[/tex]
- For [tex]\(x^3\)[/tex] terms: [tex]\(-9x^3 + 0 = -9x^3\)[/tex]
- For [tex]\(x^2\)[/tex] terms: [tex]\(0 + 4x^2 = 4x^2\)[/tex]
- For [tex]\(x\)[/tex] terms: [tex]\(7x - 3x = 4x\)[/tex]
- Constant terms: [tex]\(-1 + 2 = 1\)[/tex]
Resulting expression after combining:
[tex]\[
-3x^4 - 9x^3 + 4x^2 + 4x + 1
\][/tex]
2. Simplify the third part by expanding:
Expand [tex]\((-4x^3 + 5x - 1)(2x - 7)\)[/tex].
- Multiply [tex]\(-4x^3\)[/tex] by each term: [tex]\(-4x^3 \cdot 2x = -8x^4\)[/tex] and [tex]\(-4x^3 \cdot (-7) = 28x^3\)[/tex]
- Multiply [tex]\(5x\)[/tex] by each term: [tex]\(5x \cdot 2x = 10x^2\)[/tex] and [tex]\(5x \cdot (-7) = -35x\)[/tex]
- Multiply [tex]\(-1\)[/tex] by each term: [tex]\(-1 \cdot 2x = -2x\)[/tex] and [tex]\(-1 \cdot (-7) = 7\)[/tex]
Combine these:
[tex]\[
-8x^4 + 28x^3 + 10x^2 - 35x - 2x + 7
\][/tex]
Simplify by combining like terms:
[tex]\[
-8x^4 + 28x^3 + 10x^2 - 37x + 7
\][/tex]
3. Subtract the expanded polynomial:
Now, subtract the combined result from step 1 by the expanded expression from step 2:
[tex]\[
(-3x^4 - 9x^3 + 4x^2 + 4x + 1) - (-8x^4 + 28x^3 + 10x^2 - 37x + 7)
\][/tex]
Performing the subtraction:
- For [tex]\(x^4\)[/tex] terms: [tex]\(-3x^4 - (-8x^4) = 5x^4\)[/tex]
- For [tex]\(x^3\)[/tex] terms: [tex]\(-9x^3 - 28x^3 = -37x^3\)[/tex]
- For [tex]\(x^2\)[/tex] terms: [tex]\(4x^2 - 10x^2 = -6x^2\)[/tex]
- For [tex]\(x\)[/tex] terms: [tex]\(4x - (-37x) = 41x\)[/tex]
- Constant terms: [tex]\(1 - 7 = -6\)[/tex]
4. Final Result:
The simplified polynomial expression is:
[tex]\[
5x^4 - 37x^3 - 6x^2 + 41x - 6
\][/tex]
With this simplification, the correct answer is option C.
We have the expression:
[tex]\[
(5x^4 - 9x^3 + 7x - 1) + (-8x^4 + 4x^2 - 3x + 2) - \left((-4x^3 + 5x - 1)(2x - 7)\right)
\][/tex]
1. Combine like terms from the first two polynomials:
[tex]\[
(5x^4 - 9x^3 + 7x - 1) + (-8x^4 + 4x^2 - 3x + 2)
\][/tex]
Combine these:
- For [tex]\(x^4\)[/tex] terms: [tex]\(5x^4 - 8x^4 = -3x^4\)[/tex]
- For [tex]\(x^3\)[/tex] terms: [tex]\(-9x^3 + 0 = -9x^3\)[/tex]
- For [tex]\(x^2\)[/tex] terms: [tex]\(0 + 4x^2 = 4x^2\)[/tex]
- For [tex]\(x\)[/tex] terms: [tex]\(7x - 3x = 4x\)[/tex]
- Constant terms: [tex]\(-1 + 2 = 1\)[/tex]
Resulting expression after combining:
[tex]\[
-3x^4 - 9x^3 + 4x^2 + 4x + 1
\][/tex]
2. Simplify the third part by expanding:
Expand [tex]\((-4x^3 + 5x - 1)(2x - 7)\)[/tex].
- Multiply [tex]\(-4x^3\)[/tex] by each term: [tex]\(-4x^3 \cdot 2x = -8x^4\)[/tex] and [tex]\(-4x^3 \cdot (-7) = 28x^3\)[/tex]
- Multiply [tex]\(5x\)[/tex] by each term: [tex]\(5x \cdot 2x = 10x^2\)[/tex] and [tex]\(5x \cdot (-7) = -35x\)[/tex]
- Multiply [tex]\(-1\)[/tex] by each term: [tex]\(-1 \cdot 2x = -2x\)[/tex] and [tex]\(-1 \cdot (-7) = 7\)[/tex]
Combine these:
[tex]\[
-8x^4 + 28x^3 + 10x^2 - 35x - 2x + 7
\][/tex]
Simplify by combining like terms:
[tex]\[
-8x^4 + 28x^3 + 10x^2 - 37x + 7
\][/tex]
3. Subtract the expanded polynomial:
Now, subtract the combined result from step 1 by the expanded expression from step 2:
[tex]\[
(-3x^4 - 9x^3 + 4x^2 + 4x + 1) - (-8x^4 + 28x^3 + 10x^2 - 37x + 7)
\][/tex]
Performing the subtraction:
- For [tex]\(x^4\)[/tex] terms: [tex]\(-3x^4 - (-8x^4) = 5x^4\)[/tex]
- For [tex]\(x^3\)[/tex] terms: [tex]\(-9x^3 - 28x^3 = -37x^3\)[/tex]
- For [tex]\(x^2\)[/tex] terms: [tex]\(4x^2 - 10x^2 = -6x^2\)[/tex]
- For [tex]\(x\)[/tex] terms: [tex]\(4x - (-37x) = 41x\)[/tex]
- Constant terms: [tex]\(1 - 7 = -6\)[/tex]
4. Final Result:
The simplified polynomial expression is:
[tex]\[
5x^4 - 37x^3 - 6x^2 + 41x - 6
\][/tex]
With this simplification, the correct answer is option C.
