Answer :
Sure! Let's break down the steps to solve this expression:
Start with the given expression:
[tex]\[
(5 x^4 - 9 x^3 + 7 x - 1) + (-8 x^4 + 4 x^2 - 3 x + 2) - ((-4 x^3 + 5 x - 1)(2 x - 7))
\][/tex]
Step 1: Add the first two polynomials
- The first polynomial is [tex]\(5 x^4 - 9 x^3 + 7 x - 1\)[/tex].
- The second polynomial is [tex]\(-8 x^4 + 4 x^2 - 3 x + 2\)[/tex].
Combine like terms:
- [tex]\(5 x^4 - 8 x^4 = -3 x^4\)[/tex]
- [tex]\(-9 x^3\)[/tex] (no like term, so it remains the same)
- [tex]\(4 x^2\)[/tex] (no like term, so it remains the same)
- [tex]\(7 x - 3 x = 4 x\)[/tex]
- [tex]\(-1 + 2 = 1\)[/tex]
So the result from adding the two polynomials is:
[tex]\[-3 x^4 - 9 x^3 + 4 x^2 + 4 x + 1\][/tex]
Step 2: Expand and simplify the third polynomial
Now, consider the third polynomial:
[tex]\((-4 x^3 + 5 x - 1)(2 x - 7)\)[/tex]
Distribute each term in the first polynomial by each term in the second polynomial:
[tex]\[
= (-4 x^3) \times (2 x) + (-4 x^3) \times (-7) + (5 x) \times (2 x) + (5 x) \times (-7) + (-1) \times (2 x) + (-1) \times (-7)
\][/tex]
Calculate step-by-step:
1. [tex]\((-4 x^3) \times (2 x) = -8 x^4\)[/tex]
2. [tex]\((-4 x^3) \times (-7) = 28 x^3\)[/tex]
3. [tex]\((5 x) \times (2 x) = 10 x^2\)[/tex]
4. [tex]\((5 x) \times (-7) = -35 x\)[/tex]
5. [tex]\((-1) \times (2 x) = -2 x\)[/tex]
6. [tex]\((-1) \times (-7) = 7\)[/tex]
Combine these results:
[tex]\[-8 x^4 + 28 x^3 + 10 x^2 - 37 x + 7\][/tex]
Step 3: Subtract the expanded expression from the combined expression
Now subtract the expanded polynomial from the simplified sum of the first two polynomials:
[tex]\[
(-3 x^4 - 9 x^3 + 4 x^2 + 4 x + 1) - (-8 x^4 + 28 x^3 + 10 x^2 - 37 x + 7)
\][/tex]
When subtracting, distribute the negative sign:
[tex]\[
-3 x^4 - 9 x^3 + 4 x^2 + 4 x + 1 + 8 x^4 - 28 x^3 - 10 x^2 + 37 x - 7
\][/tex]
Combine like terms:
- [tex]\(-3 x^4 + 8 x^4 = 5 x^4\)[/tex]
- [tex]\(-9 x^3 - 28 x^3 = -37 x^3\)[/tex]
- [tex]\(4 x^2 - 10 x^2 = -6 x^2\)[/tex]
- [tex]\(4 x + 37 x = 41 x\)[/tex]
- [tex]\(1 - 7 = -6\)[/tex]
This results in the final polynomial:
[tex]\[5 x^4 - 37 x^3 - 6 x^2 + 41 x - 6\][/tex]
Thus, the correct answer is:
C. [tex]\(5 x^4 - 37 x^3 - 6 x^2 + 41 x - 6\)[/tex]
Start with the given expression:
[tex]\[
(5 x^4 - 9 x^3 + 7 x - 1) + (-8 x^4 + 4 x^2 - 3 x + 2) - ((-4 x^3 + 5 x - 1)(2 x - 7))
\][/tex]
Step 1: Add the first two polynomials
- The first polynomial is [tex]\(5 x^4 - 9 x^3 + 7 x - 1\)[/tex].
- The second polynomial is [tex]\(-8 x^4 + 4 x^2 - 3 x + 2\)[/tex].
Combine like terms:
- [tex]\(5 x^4 - 8 x^4 = -3 x^4\)[/tex]
- [tex]\(-9 x^3\)[/tex] (no like term, so it remains the same)
- [tex]\(4 x^2\)[/tex] (no like term, so it remains the same)
- [tex]\(7 x - 3 x = 4 x\)[/tex]
- [tex]\(-1 + 2 = 1\)[/tex]
So the result from adding the two polynomials is:
[tex]\[-3 x^4 - 9 x^3 + 4 x^2 + 4 x + 1\][/tex]
Step 2: Expand and simplify the third polynomial
Now, consider the third polynomial:
[tex]\((-4 x^3 + 5 x - 1)(2 x - 7)\)[/tex]
Distribute each term in the first polynomial by each term in the second polynomial:
[tex]\[
= (-4 x^3) \times (2 x) + (-4 x^3) \times (-7) + (5 x) \times (2 x) + (5 x) \times (-7) + (-1) \times (2 x) + (-1) \times (-7)
\][/tex]
Calculate step-by-step:
1. [tex]\((-4 x^3) \times (2 x) = -8 x^4\)[/tex]
2. [tex]\((-4 x^3) \times (-7) = 28 x^3\)[/tex]
3. [tex]\((5 x) \times (2 x) = 10 x^2\)[/tex]
4. [tex]\((5 x) \times (-7) = -35 x\)[/tex]
5. [tex]\((-1) \times (2 x) = -2 x\)[/tex]
6. [tex]\((-1) \times (-7) = 7\)[/tex]
Combine these results:
[tex]\[-8 x^4 + 28 x^3 + 10 x^2 - 37 x + 7\][/tex]
Step 3: Subtract the expanded expression from the combined expression
Now subtract the expanded polynomial from the simplified sum of the first two polynomials:
[tex]\[
(-3 x^4 - 9 x^3 + 4 x^2 + 4 x + 1) - (-8 x^4 + 28 x^3 + 10 x^2 - 37 x + 7)
\][/tex]
When subtracting, distribute the negative sign:
[tex]\[
-3 x^4 - 9 x^3 + 4 x^2 + 4 x + 1 + 8 x^4 - 28 x^3 - 10 x^2 + 37 x - 7
\][/tex]
Combine like terms:
- [tex]\(-3 x^4 + 8 x^4 = 5 x^4\)[/tex]
- [tex]\(-9 x^3 - 28 x^3 = -37 x^3\)[/tex]
- [tex]\(4 x^2 - 10 x^2 = -6 x^2\)[/tex]
- [tex]\(4 x + 37 x = 41 x\)[/tex]
- [tex]\(1 - 7 = -6\)[/tex]
This results in the final polynomial:
[tex]\[5 x^4 - 37 x^3 - 6 x^2 + 41 x - 6\][/tex]
Thus, the correct answer is:
C. [tex]\(5 x^4 - 37 x^3 - 6 x^2 + 41 x - 6\)[/tex]
