Answer :
Let's simplify the given polynomial expression step-by-step.
We have the expression:
[tex]\[
(5x^4 - 9x^3 + 7x - 1) + (-8x^4 + 4x^2 - 3x + 2) - ((-4x^3 + 5x - 1)(2x - 7))
\][/tex]
Step 1: Combine the first two polynomials
First, add the first two polynomial expressions:
[tex]\[
(5x^4 - 9x^3 + 7x - 1) + (-8x^4 + 4x^2 - 3x + 2)
\][/tex]
Combine like terms:
- For [tex]\(x^4\)[/tex]: [tex]\(5x^4 + (-8x^4) = -3x^4\)[/tex]
- For [tex]\(x^3\)[/tex]: [tex]\(-9x^3 + 0 = -9x^3\)[/tex]
- For [tex]\(x^2\)[/tex]: [tex]\(0 + 4x^2 = 4x^2\)[/tex]
- For [tex]\(x\)[/tex]: [tex]\(7x + (-3x) = 4x\)[/tex]
- For the constant terms: [tex]\(-1 + 2 = 1\)[/tex]
So, we have:
[tex]\[
-3x^4 - 9x^3 + 4x^2 + 4x + 1
\][/tex]
Step 2: Simplify the third polynomial part
Now, simplify [tex]\((-4x^3 + 5x - 1)(2x - 7)\)[/tex] by distributing:
[tex]\[
(-4x^3 + 5x - 1) \times (2x - 7)
\][/tex]
Distribute each term in [tex]\((-4x^3 + 5x - 1)\)[/tex]:
1. [tex]\(-4x^3 \times 2x = -8x^4\)[/tex]
2. [tex]\(-4x^3 \times (-7) = 28x^3\)[/tex]
3. [tex]\(5x \times 2x = 10x^2\)[/tex]
4. [tex]\(5x \times (-7) = -35x\)[/tex]
5. [tex]\(-1 \times 2x = -2x\)[/tex]
6. [tex]\(-1 \times (-7) = 7\)[/tex]
Combine these results:
[tex]\[
-8x^4 + 28x^3 + 10x^2 - 35x - 2x + 7
\][/tex]
Which simplifies to:
[tex]\[
-8x^4 + 28x^3 + 10x^2 - 37x + 7
\][/tex]
Step 3: Subtract the expressions
Now subtract the result of the third polynomial expression from the sum of the first two:
[tex]\[
(-3x^4 - 9x^3 + 4x^2 + 4x + 1) - (-8x^4 + 28x^3 + 10x^2 - 37x + 7)
\][/tex]
Distribute the negative sign:
[tex]\[
-3x^4 - 9x^3 + 4x^2 + 4x + 1 + 8x^4 - 28x^3 - 10x^2 + 37x - 7
\][/tex]
Combine like terms:
- For [tex]\(x^4\)[/tex]: [tex]\(-3x^4 + 8x^4 = 5x^4\)[/tex]
- For [tex]\(x^3\)[/tex]: [tex]\(-9x^3 - 28x^3 = -37x^3\)[/tex]
- For [tex]\(x^2\)[/tex]: [tex]\(4x^2 - 10x^2 = -6x^2\)[/tex]
- For [tex]\(x\)[/tex]: [tex]\(4x + 37x = 41x\)[/tex]
- For the constant terms: [tex]\(1 - 7 = -6\)[/tex]
Thus, the simplified expression is:
[tex]\[
5x^4 - 37x^3 - 6x^2 + 41x - 6
\][/tex]
So, the correct answer is A. [tex]\(5x^4 - 37x^3 - 6x^2 + 41x - 6\)[/tex].
We have the expression:
[tex]\[
(5x^4 - 9x^3 + 7x - 1) + (-8x^4 + 4x^2 - 3x + 2) - ((-4x^3 + 5x - 1)(2x - 7))
\][/tex]
Step 1: Combine the first two polynomials
First, add the first two polynomial expressions:
[tex]\[
(5x^4 - 9x^3 + 7x - 1) + (-8x^4 + 4x^2 - 3x + 2)
\][/tex]
Combine like terms:
- For [tex]\(x^4\)[/tex]: [tex]\(5x^4 + (-8x^4) = -3x^4\)[/tex]
- For [tex]\(x^3\)[/tex]: [tex]\(-9x^3 + 0 = -9x^3\)[/tex]
- For [tex]\(x^2\)[/tex]: [tex]\(0 + 4x^2 = 4x^2\)[/tex]
- For [tex]\(x\)[/tex]: [tex]\(7x + (-3x) = 4x\)[/tex]
- For the constant terms: [tex]\(-1 + 2 = 1\)[/tex]
So, we have:
[tex]\[
-3x^4 - 9x^3 + 4x^2 + 4x + 1
\][/tex]
Step 2: Simplify the third polynomial part
Now, simplify [tex]\((-4x^3 + 5x - 1)(2x - 7)\)[/tex] by distributing:
[tex]\[
(-4x^3 + 5x - 1) \times (2x - 7)
\][/tex]
Distribute each term in [tex]\((-4x^3 + 5x - 1)\)[/tex]:
1. [tex]\(-4x^3 \times 2x = -8x^4\)[/tex]
2. [tex]\(-4x^3 \times (-7) = 28x^3\)[/tex]
3. [tex]\(5x \times 2x = 10x^2\)[/tex]
4. [tex]\(5x \times (-7) = -35x\)[/tex]
5. [tex]\(-1 \times 2x = -2x\)[/tex]
6. [tex]\(-1 \times (-7) = 7\)[/tex]
Combine these results:
[tex]\[
-8x^4 + 28x^3 + 10x^2 - 35x - 2x + 7
\][/tex]
Which simplifies to:
[tex]\[
-8x^4 + 28x^3 + 10x^2 - 37x + 7
\][/tex]
Step 3: Subtract the expressions
Now subtract the result of the third polynomial expression from the sum of the first two:
[tex]\[
(-3x^4 - 9x^3 + 4x^2 + 4x + 1) - (-8x^4 + 28x^3 + 10x^2 - 37x + 7)
\][/tex]
Distribute the negative sign:
[tex]\[
-3x^4 - 9x^3 + 4x^2 + 4x + 1 + 8x^4 - 28x^3 - 10x^2 + 37x - 7
\][/tex]
Combine like terms:
- For [tex]\(x^4\)[/tex]: [tex]\(-3x^4 + 8x^4 = 5x^4\)[/tex]
- For [tex]\(x^3\)[/tex]: [tex]\(-9x^3 - 28x^3 = -37x^3\)[/tex]
- For [tex]\(x^2\)[/tex]: [tex]\(4x^2 - 10x^2 = -6x^2\)[/tex]
- For [tex]\(x\)[/tex]: [tex]\(4x + 37x = 41x\)[/tex]
- For the constant terms: [tex]\(1 - 7 = -6\)[/tex]
Thus, the simplified expression is:
[tex]\[
5x^4 - 37x^3 - 6x^2 + 41x - 6
\][/tex]
So, the correct answer is A. [tex]\(5x^4 - 37x^3 - 6x^2 + 41x - 6\)[/tex].
