Answer :
To simplify the given polynomial expression, we need to follow a series of steps. Let's break it down:
1. Identify the expressions:
We have three polynomial expressions that we need to work with:
- [tex]\( (5x^4 - 9x^3 + 7x - 1) \)[/tex]
- [tex]\( (-8x^4 + 4x^2 - 3x + 2) \)[/tex]
- [tex]\( (-4x^3 + 5x - 1)(2x - 7) \)[/tex]
2. Simplify by handling operations in correct order:
- Multiply the third expression:
We need to distribute the terms in the expression [tex]\((-4x^3 + 5x - 1)(2x - 7)\)[/tex] using the distributive property (FOIL method):
[tex]\[
\begin{align*}
(-4x^3 + 5x - 1)(2x - 7) &= (-4x^3)(2x) + (-4x^3)(-7) + (5x)(2x) + (5x)(-7) + (-1)(2x) + (-1)(-7) \\
&= -8x^4 + 28x^3 + 10x^2 - 35x - 2x + 7
\end{align*}
\][/tex]
Combine like terms:
[tex]\[
-8x^4 + 28x^3 + 10x^2 - 37x + 7
\][/tex]
- Add/Subtract the expressions:
Next, we combine the original expressions, making sure to subtract the expanded third expression from the sum of the first two:
[tex]\[
\left(5x^4 - 9x^3 + 7x - 1\right) + \left(-8x^4 + 4x^2 - 3x + 2\right) - \left(-8x^4 + 28x^3 + 10x^2 - 37x + 7\right)
\][/tex]
3. Combine the expressions:
Let's now combine the expressions by adding and subtracting like terms:
- Combine the [tex]\(x^4\)[/tex] terms: [tex]\(5x^4 - 8x^4 + 8x^4 = 5x^4\)[/tex]
- Combine the [tex]\(x^3\)[/tex] terms: [tex]\(-9x^3 - 28x^3 = -37x^3\)[/tex]
- Combine the [tex]\(x^2\)[/tex] terms: [tex]\(4x^2 - 10x^2 = -6x^2\)[/tex]
- Combine the [tex]\(x\)[/tex] terms: [tex]\(7x - 3x + 37x = 41x\)[/tex]
- Combine the constant terms: [tex]\(-1 + 2 - 7 = -6\)[/tex]
The simplified expression is:
[tex]\[
5x^4 - 37x^3 - 6x^2 + 41x - 6
\][/tex]
4. Selecting the correct answer:
The expression [tex]\(5x^4 - 37x^3 - 6x^2 + 41x - 6\)[/tex] matches option C.
Therefore, the correct simplified expression is:
C. [tex]\(5x^4 - 37x^3 - 6x^2 + 41x - 6\)[/tex]
1. Identify the expressions:
We have three polynomial expressions that we need to work with:
- [tex]\( (5x^4 - 9x^3 + 7x - 1) \)[/tex]
- [tex]\( (-8x^4 + 4x^2 - 3x + 2) \)[/tex]
- [tex]\( (-4x^3 + 5x - 1)(2x - 7) \)[/tex]
2. Simplify by handling operations in correct order:
- Multiply the third expression:
We need to distribute the terms in the expression [tex]\((-4x^3 + 5x - 1)(2x - 7)\)[/tex] using the distributive property (FOIL method):
[tex]\[
\begin{align*}
(-4x^3 + 5x - 1)(2x - 7) &= (-4x^3)(2x) + (-4x^3)(-7) + (5x)(2x) + (5x)(-7) + (-1)(2x) + (-1)(-7) \\
&= -8x^4 + 28x^3 + 10x^2 - 35x - 2x + 7
\end{align*}
\][/tex]
Combine like terms:
[tex]\[
-8x^4 + 28x^3 + 10x^2 - 37x + 7
\][/tex]
- Add/Subtract the expressions:
Next, we combine the original expressions, making sure to subtract the expanded third expression from the sum of the first two:
[tex]\[
\left(5x^4 - 9x^3 + 7x - 1\right) + \left(-8x^4 + 4x^2 - 3x + 2\right) - \left(-8x^4 + 28x^3 + 10x^2 - 37x + 7\right)
\][/tex]
3. Combine the expressions:
Let's now combine the expressions by adding and subtracting like terms:
- Combine the [tex]\(x^4\)[/tex] terms: [tex]\(5x^4 - 8x^4 + 8x^4 = 5x^4\)[/tex]
- Combine the [tex]\(x^3\)[/tex] terms: [tex]\(-9x^3 - 28x^3 = -37x^3\)[/tex]
- Combine the [tex]\(x^2\)[/tex] terms: [tex]\(4x^2 - 10x^2 = -6x^2\)[/tex]
- Combine the [tex]\(x\)[/tex] terms: [tex]\(7x - 3x + 37x = 41x\)[/tex]
- Combine the constant terms: [tex]\(-1 + 2 - 7 = -6\)[/tex]
The simplified expression is:
[tex]\[
5x^4 - 37x^3 - 6x^2 + 41x - 6
\][/tex]
4. Selecting the correct answer:
The expression [tex]\(5x^4 - 37x^3 - 6x^2 + 41x - 6\)[/tex] matches option C.
Therefore, the correct simplified expression is:
C. [tex]\(5x^4 - 37x^3 - 6x^2 + 41x - 6\)[/tex]
