Answer :
Let's simplify the given polynomial expression step-by-step:
We have three expressions:
1. [tex]\((5x^4 - 9x^3 + 7x - 1)\)[/tex]
2. [tex]\((-8x^4 + 4x^2 - 3x + 2)\)[/tex]
3. [tex]\((-4x^3 + 5x - 1)(2x - 7)\)[/tex]
The expression we need to simplify is:
[tex]\[
(5x^4 - 9x^3 + 7x - 1) + (-8x^4 + 4x^2 - 3x + 2) - ((-4x^3 + 5x - 1)(2x - 7))
\][/tex]
Step 1: Simplify the third expression by expanding it.
Let's first expand [tex]\((-4x^3 + 5x - 1)(2x - 7)\)[/tex]:
- Multiply each term in the first polynomial by each term in the second polynomial.
[tex]\[
\begin{align*}
(-4x^3)(2x) & = -8x^4, \\
(-4x^3)(-7) & = 28x^3, \\
(5x)(2x) & = 10x^2, \\
(5x)(-7) & = -35x, \\
(-1)(2x) & = -2x, \\
(-1)(-7) & = 7.
\end{align*}
\][/tex]
Combine these results:
[tex]\[ -8x^4 + 28x^3 + 10x^2 - 35x - 2x + 7 \][/tex]
[tex]\[ = -8x^4 + 28x^3 + 10x^2 - 37x + 7 \][/tex]
Step 2: Substitute and simplify the whole expression.
Substitute the expanded expression back into the main expression:
[tex]\[
(5x^4 - 9x^3 + 7x - 1) + (-8x^4 + 4x^2 - 3x + 2) - (-8x^4 + 28x^3 + 10x^2 - 37x + 7)
\][/tex]
Step 3: 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]\(-9x^3 - 28x^3 = -37x^3\)[/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]\(-1 - 2 = -3\)[/tex].
The simplified polynomial expression is:
[tex]\[
5x^4 - 37x^3 - 6x^2 + 41x - 6
\][/tex]
So, the correct answer is B. [tex]\(5x^4 - 37x^3 - 6x^2 + 41x - 6\)[/tex].
We have three expressions:
1. [tex]\((5x^4 - 9x^3 + 7x - 1)\)[/tex]
2. [tex]\((-8x^4 + 4x^2 - 3x + 2)\)[/tex]
3. [tex]\((-4x^3 + 5x - 1)(2x - 7)\)[/tex]
The expression we need to simplify is:
[tex]\[
(5x^4 - 9x^3 + 7x - 1) + (-8x^4 + 4x^2 - 3x + 2) - ((-4x^3 + 5x - 1)(2x - 7))
\][/tex]
Step 1: Simplify the third expression by expanding it.
Let's first expand [tex]\((-4x^3 + 5x - 1)(2x - 7)\)[/tex]:
- Multiply each term in the first polynomial by each term in the second polynomial.
[tex]\[
\begin{align*}
(-4x^3)(2x) & = -8x^4, \\
(-4x^3)(-7) & = 28x^3, \\
(5x)(2x) & = 10x^2, \\
(5x)(-7) & = -35x, \\
(-1)(2x) & = -2x, \\
(-1)(-7) & = 7.
\end{align*}
\][/tex]
Combine these results:
[tex]\[ -8x^4 + 28x^3 + 10x^2 - 35x - 2x + 7 \][/tex]
[tex]\[ = -8x^4 + 28x^3 + 10x^2 - 37x + 7 \][/tex]
Step 2: Substitute and simplify the whole expression.
Substitute the expanded expression back into the main expression:
[tex]\[
(5x^4 - 9x^3 + 7x - 1) + (-8x^4 + 4x^2 - 3x + 2) - (-8x^4 + 28x^3 + 10x^2 - 37x + 7)
\][/tex]
Step 3: 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]\(-9x^3 - 28x^3 = -37x^3\)[/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]\(-1 - 2 = -3\)[/tex].
The simplified polynomial expression is:
[tex]\[
5x^4 - 37x^3 - 6x^2 + 41x - 6
\][/tex]
So, the correct answer is B. [tex]\(5x^4 - 37x^3 - 6x^2 + 41x - 6\)[/tex].
