Answer :
Sure! Let's work through the problem step by step to simplify the polynomial expression:
We are given the expression:
[tex]\[
(5x^4 - 9x^3 + 7x - 1) + (-8x^4 + 4x^2 - 3x + 2) - \left((-4x^3 + 5x - 1)(2x - 7)\right)
\][/tex]
Step 1: Simplify inside the parentheses
First, we'll perform the multiplication inside the third set of parentheses:
[tex]\[
(-4x^3 + 5x - 1)(2x - 7)
\][/tex]
Using the distributive property:
1. [tex]\(-4x^3 \cdot 2x = -8x^4\)[/tex]
2. [tex]\(-4x^3 \cdot (-7) = 28x^3\)[/tex]
3. [tex]\(5x \cdot 2x = 10x^2\)[/tex]
4. [tex]\(5x \cdot (-7) = -35x\)[/tex]
5. [tex]\(-1 \cdot 2x = -2x\)[/tex]
6. [tex]\(-1 \cdot (-7) = 7\)[/tex]
Combine these results:
[tex]\[
-8x^4 + 28x^3 + 10x^2 - 35x - 2x + 7
\][/tex]
Combine like terms:
[tex]\[
-8x^4 + 28x^3 + 10x^2 - 37x + 7
\][/tex]
Step 2: Plug the simplified expression back in
Now substitute this simplified expression back into the original problem:
[tex]\[
(5x^4 - 9x^3 + 7x - 1) + (-8x^4 + 4x^2 - 3x + 2) - (-8x^4 + 28x^3 + 10x^2 - 37x + 7)
\][/tex]
Step 3: Distribute the negative sign
Distribute the negative sign through the third expression:
[tex]\[
5x^4 - 9x^3 + 7x - 1 - 8x^4 + 4x^2 - 3x + 2 + 8x^4 - 28x^3 - 10x^2 + 37x - 7
\][/tex]
Step 4: Combine like terms
Let's combine the like terms:
- [tex]\(x^4\)[/tex] terms: [tex]\(5x^4 - 8x^4 + 8x^4 = 5x^4\)[/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]\(7x - 3x + 37x = 41x\)[/tex]
- Constant terms: [tex]\(-1 + 2 - 7 = -6\)[/tex]
Putting it all together, we have:
[tex]\[
5x^4 - 37x^3 - 6x^2 + 41x - 6
\][/tex]
So, the simplified polynomial expression is:
[tex]\[
5x^4 - 37x^3 - 6x^2 + 41x - 6
\][/tex]
Therefore, the correct answer is B. [tex]\(5x^4 - 37x^3 - 6x^2 + 41x - 6\)[/tex].
We are given the expression:
[tex]\[
(5x^4 - 9x^3 + 7x - 1) + (-8x^4 + 4x^2 - 3x + 2) - \left((-4x^3 + 5x - 1)(2x - 7)\right)
\][/tex]
Step 1: Simplify inside the parentheses
First, we'll perform the multiplication inside the third set of parentheses:
[tex]\[
(-4x^3 + 5x - 1)(2x - 7)
\][/tex]
Using the distributive property:
1. [tex]\(-4x^3 \cdot 2x = -8x^4\)[/tex]
2. [tex]\(-4x^3 \cdot (-7) = 28x^3\)[/tex]
3. [tex]\(5x \cdot 2x = 10x^2\)[/tex]
4. [tex]\(5x \cdot (-7) = -35x\)[/tex]
5. [tex]\(-1 \cdot 2x = -2x\)[/tex]
6. [tex]\(-1 \cdot (-7) = 7\)[/tex]
Combine these results:
[tex]\[
-8x^4 + 28x^3 + 10x^2 - 35x - 2x + 7
\][/tex]
Combine like terms:
[tex]\[
-8x^4 + 28x^3 + 10x^2 - 37x + 7
\][/tex]
Step 2: Plug the simplified expression back in
Now substitute this simplified expression back into the original problem:
[tex]\[
(5x^4 - 9x^3 + 7x - 1) + (-8x^4 + 4x^2 - 3x + 2) - (-8x^4 + 28x^3 + 10x^2 - 37x + 7)
\][/tex]
Step 3: Distribute the negative sign
Distribute the negative sign through the third expression:
[tex]\[
5x^4 - 9x^3 + 7x - 1 - 8x^4 + 4x^2 - 3x + 2 + 8x^4 - 28x^3 - 10x^2 + 37x - 7
\][/tex]
Step 4: Combine like terms
Let's combine the like terms:
- [tex]\(x^4\)[/tex] terms: [tex]\(5x^4 - 8x^4 + 8x^4 = 5x^4\)[/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]\(7x - 3x + 37x = 41x\)[/tex]
- Constant terms: [tex]\(-1 + 2 - 7 = -6\)[/tex]
Putting it all together, we have:
[tex]\[
5x^4 - 37x^3 - 6x^2 + 41x - 6
\][/tex]
So, the simplified polynomial expression is:
[tex]\[
5x^4 - 37x^3 - 6x^2 + 41x - 6
\][/tex]
Therefore, the correct answer is B. [tex]\(5x^4 - 37x^3 - 6x^2 + 41x - 6\)[/tex].
