Answer :
Let's simplify the given polynomial expression step by step:
The expression is:
[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
Add the polynomials [tex]\(5x^4 - 9x^3 + 7x - 1\)[/tex] and [tex]\(-8x^4 + 4x^2 - 3x + 2\)[/tex]:
1. Combine the [tex]\(x^4\)[/tex] terms: [tex]\(5x^4 + (-8x^4) = -3x^4\)[/tex]
2. Combine the [tex]\(x^3\)[/tex] terms: [tex]\(-9x^3\)[/tex]
3. Combine the [tex]\(x^2\)[/tex] terms: [tex]\(4x^2\)[/tex]
4. Combine the [tex]\(x\)[/tex] terms: [tex]\(7x - 3x = 4x\)[/tex]
5. Combine the constant terms: [tex]\(-1 + 2 = 1\)[/tex]
After combining the like terms, the expression becomes:
[tex]\[-3x^4 - 9x^3 + 4x^2 + 4x + 1\][/tex]
### Step 2: Distribute in the third polynomial
Now, we need to multiply [tex]\((-4x^3 + 5x - 1)\)[/tex] by [tex]\((2x - 7)\)[/tex].
Perform the multiplication:
1. Multiply [tex]\(-4x^3\)[/tex] by each term in [tex]\(2x - 7\)[/tex]:
- [tex]\(-4x^3 \cdot 2x = -8x^4\)[/tex]
- [tex]\(-4x^3 \cdot (-7) = 28x^3\)[/tex]
2. Multiply [tex]\(5x\)[/tex] by each term in [tex]\(2x - 7\)[/tex]:
- [tex]\(5x \cdot 2x = 10x^2\)[/tex]
- [tex]\(5x \cdot (-7) = -35x\)[/tex]
3. Multiply [tex]\(-1\)[/tex] by each term in [tex]\(2x - 7\)[/tex]:
- [tex]\(-1 \cdot 2x = -2x\)[/tex]
- [tex]\(-1 \cdot (-7) = 7\)[/tex]
Combine all these results:
[tex]\(-8x^4 + 28x^3 + 10x^2 - 35x - 2x + 7\)[/tex]
Simplify this by combining like terms:
[tex]\(-8x^4 + 28x^3 + 10x^2 - 37x + 7\)[/tex]
### Step 3: Subtract the result from the first expression
Now subtract the result of the multiplication from the expression we got in Step 1:
[tex]\[
(-3x^4 - 9x^3 + 4x^2 + 4x + 1) - (-8x^4 + 28x^3 + 10x^2 - 37x + 7)
\][/tex]
1. Subtract the [tex]\(x^4\)[/tex] terms: [tex]\(-3x^4 - (-8x^4) = 5x^4\)[/tex]
2. Subtract the [tex]\(x^3\)[/tex] terms: [tex]\(-9x^3 - 28x^3 = -37x^3\)[/tex]
3. Subtract the [tex]\(x^2\)[/tex] terms: [tex]\(4x^2 - 10x^2 = -6x^2\)[/tex]
4. Subtract the [tex]\(x\)[/tex] terms: [tex]\(4x - (-37x) = 41x\)[/tex]
5. Subtract the constants: [tex]\(1 - 7 = -6\)[/tex]
The final simplified expression is:
[tex]\[5x^4 - 37x^3 - 6x^2 + 41x - 6\][/tex]
Thus, the correct answer is:
A. [tex]\(5x^4 - 37x^3 - 6x^2 + 41x - 6\)[/tex]
The expression is:
[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
Add the polynomials [tex]\(5x^4 - 9x^3 + 7x - 1\)[/tex] and [tex]\(-8x^4 + 4x^2 - 3x + 2\)[/tex]:
1. Combine the [tex]\(x^4\)[/tex] terms: [tex]\(5x^4 + (-8x^4) = -3x^4\)[/tex]
2. Combine the [tex]\(x^3\)[/tex] terms: [tex]\(-9x^3\)[/tex]
3. Combine the [tex]\(x^2\)[/tex] terms: [tex]\(4x^2\)[/tex]
4. Combine the [tex]\(x\)[/tex] terms: [tex]\(7x - 3x = 4x\)[/tex]
5. Combine the constant terms: [tex]\(-1 + 2 = 1\)[/tex]
After combining the like terms, the expression becomes:
[tex]\[-3x^4 - 9x^3 + 4x^2 + 4x + 1\][/tex]
### Step 2: Distribute in the third polynomial
Now, we need to multiply [tex]\((-4x^3 + 5x - 1)\)[/tex] by [tex]\((2x - 7)\)[/tex].
Perform the multiplication:
1. Multiply [tex]\(-4x^3\)[/tex] by each term in [tex]\(2x - 7\)[/tex]:
- [tex]\(-4x^3 \cdot 2x = -8x^4\)[/tex]
- [tex]\(-4x^3 \cdot (-7) = 28x^3\)[/tex]
2. Multiply [tex]\(5x\)[/tex] by each term in [tex]\(2x - 7\)[/tex]:
- [tex]\(5x \cdot 2x = 10x^2\)[/tex]
- [tex]\(5x \cdot (-7) = -35x\)[/tex]
3. Multiply [tex]\(-1\)[/tex] by each term in [tex]\(2x - 7\)[/tex]:
- [tex]\(-1 \cdot 2x = -2x\)[/tex]
- [tex]\(-1 \cdot (-7) = 7\)[/tex]
Combine all these results:
[tex]\(-8x^4 + 28x^3 + 10x^2 - 35x - 2x + 7\)[/tex]
Simplify this by combining like terms:
[tex]\(-8x^4 + 28x^3 + 10x^2 - 37x + 7\)[/tex]
### Step 3: Subtract the result from the first expression
Now subtract the result of the multiplication from the expression we got in Step 1:
[tex]\[
(-3x^4 - 9x^3 + 4x^2 + 4x + 1) - (-8x^4 + 28x^3 + 10x^2 - 37x + 7)
\][/tex]
1. Subtract the [tex]\(x^4\)[/tex] terms: [tex]\(-3x^4 - (-8x^4) = 5x^4\)[/tex]
2. Subtract the [tex]\(x^3\)[/tex] terms: [tex]\(-9x^3 - 28x^3 = -37x^3\)[/tex]
3. Subtract the [tex]\(x^2\)[/tex] terms: [tex]\(4x^2 - 10x^2 = -6x^2\)[/tex]
4. Subtract the [tex]\(x\)[/tex] terms: [tex]\(4x - (-37x) = 41x\)[/tex]
5. Subtract the constants: [tex]\(1 - 7 = -6\)[/tex]
The final simplified expression is:
[tex]\[5x^4 - 37x^3 - 6x^2 + 41x - 6\][/tex]
Thus, the correct answer is:
A. [tex]\(5x^4 - 37x^3 - 6x^2 + 41x - 6\)[/tex]
