Answer :
Certainly! To simplify the given polynomial expression, follow these steps:
1. Identify the parts of the expression:
We have three polynomial expressions:
- [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. Expand the product in the third polynomial:
Consider the expression [tex]\((-4x^3 + 5x - 1)(2x - 7)\)[/tex]. We'll expand it by distributing each term from the first polynomial to each term of the second polynomial:
- [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]
Combine the [tex]\(x\)[/tex] terms:
- [tex]\( -35x - 2x = -37x\)[/tex]
Now, the expanded form of [tex]\((-4x^3 + 5x - 1)(2x - 7)\)[/tex] becomes:
[tex]\(-8x^4 + 28x^3 + 10x^2 - 37x + 7\)[/tex]
3. Combine all polynomials:
Now, we will subtract the expanded polynomial from the sum of the first two polynomials:
[tex]\[
(5x^4 - 9x^3 + 7x - 1) + (-8x^4 + 4x^2 - 3x + 2) - (-8x^4 + 28x^3 + 10x^2 - 37x + 7)
\][/tex]
4. Perform the additions and subtractions:
- For the [tex]\(x^4\)[/tex] term:
[tex]\[
5x^4 - 8x^4 + 8x^4 = 5x^4
\][/tex]
- For the [tex]\(x^3\)[/tex] term:
[tex]\[
-9x^3 - 0x^3 - 28x^3 = -37x^3
\][/tex]
- For the [tex]\(x^2\)[/tex] term:
[tex]\[
0x^2 + 4x^2 - 10x^2 = -6x^2
\][/tex]
- For the [tex]\(x\)[/tex] term:
[tex]\[
7x - 3x + 37x = 41x
\][/tex]
- For the constant term:
[tex]\[
-1 + 2 - 7 = -6
\][/tex]
5. Write the simplified expression:
The simplified polynomial expression is:
[tex]\[
5x^4 - 37x^3 - 6x^2 + 41x - 6
\][/tex]
The answer matches option D.
Answer: D. [tex]\(5x^4 - 37x^3 - 6x^2 + 41x - 6\)[/tex]
1. Identify the parts of the expression:
We have three polynomial expressions:
- [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. Expand the product in the third polynomial:
Consider the expression [tex]\((-4x^3 + 5x - 1)(2x - 7)\)[/tex]. We'll expand it by distributing each term from the first polynomial to each term of the second polynomial:
- [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]
Combine the [tex]\(x\)[/tex] terms:
- [tex]\( -35x - 2x = -37x\)[/tex]
Now, the expanded form of [tex]\((-4x^3 + 5x - 1)(2x - 7)\)[/tex] becomes:
[tex]\(-8x^4 + 28x^3 + 10x^2 - 37x + 7\)[/tex]
3. Combine all polynomials:
Now, we will subtract the expanded polynomial from the sum of the first two polynomials:
[tex]\[
(5x^4 - 9x^3 + 7x - 1) + (-8x^4 + 4x^2 - 3x + 2) - (-8x^4 + 28x^3 + 10x^2 - 37x + 7)
\][/tex]
4. Perform the additions and subtractions:
- For the [tex]\(x^4\)[/tex] term:
[tex]\[
5x^4 - 8x^4 + 8x^4 = 5x^4
\][/tex]
- For the [tex]\(x^3\)[/tex] term:
[tex]\[
-9x^3 - 0x^3 - 28x^3 = -37x^3
\][/tex]
- For the [tex]\(x^2\)[/tex] term:
[tex]\[
0x^2 + 4x^2 - 10x^2 = -6x^2
\][/tex]
- For the [tex]\(x\)[/tex] term:
[tex]\[
7x - 3x + 37x = 41x
\][/tex]
- For the constant term:
[tex]\[
-1 + 2 - 7 = -6
\][/tex]
5. Write the simplified expression:
The simplified polynomial expression is:
[tex]\[
5x^4 - 37x^3 - 6x^2 + 41x - 6
\][/tex]
The answer matches option D.
Answer: D. [tex]\(5x^4 - 37x^3 - 6x^2 + 41x - 6\)[/tex]
