Answer :
Let's simplify the given polynomial expression step-by-step:
The original 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 Like Terms from the First Two Polynomials
First, we'll combine the terms in the first two polynomials:
- [tex]\( x^4 \)[/tex] terms: [tex]\( 5x^4 - 8x^4 = -3x^4 \)[/tex]
- [tex]\( x^3 \)[/tex] terms: [tex]\( -9x^3 \)[/tex] (no additional [tex]\( x^3 \)[/tex] term in the second polynomial)
- [tex]\( x^2 \)[/tex] terms: [tex]\( 4x^2 \)[/tex] (only present in the second polynomial)
- [tex]\( x \)[/tex] terms: [tex]\( 7x - 3x = 4x \)[/tex]
- Constant terms: [tex]\( -1 + 2 = 1 \)[/tex]
Combining these gives:
[tex]\[
-3x^4 - 9x^3 + 4x^2 + 4x + 1
\][/tex]
### Step 2: Simplify the Subtraction of the Last Expression
Now, we need to handle the subtraction of the last polynomial expression:
[tex]\[
- (-4x^3 + 5x - 1)(2x - 7)
\][/tex]
First, expand the expression [tex]\((-4x^3 + 5x - 1)(2x - 7)\)[/tex]:
- Multiplying [tex]\(-4x^3\)[/tex] with:
- [tex]\(2x\)[/tex] gives [tex]\(-8x^4\)[/tex]
- [tex]\(-7\)[/tex] gives [tex]\(28x^3\)[/tex]
- Multiplying [tex]\(5x\)[/tex] with:
- [tex]\(2x\)[/tex] gives [tex]\(10x^2\)[/tex]
- [tex]\(-7\)[/tex] gives [tex]\(-35x\)[/tex]
- Multiplying [tex]\(-1\)[/tex] with:
- [tex]\(2x\)[/tex] gives [tex]\(-2x\)[/tex]
- [tex]\(-7\)[/tex] gives [tex]\(7\)[/tex]
Combine the results of these products:
[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: Combine All Expressions
Now, substitute back and combine with the first part:
[tex]\[
(-3x^4 - 9x^3 + 4x^2 + 4x + 1) - (-8x^4 + 28x^3 + 10x^2 - 37x + 7)
\][/tex]
Subtract each corresponding term:
- [tex]\( x^4 \)[/tex] terms: [tex]\(-3x^4 - (-8x^4) = 5x^4\)[/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]\(4x - (-37x) = 41x\)[/tex]
- Constant terms: [tex]\(1 - 7 = -6\)[/tex]
Bringing it all together, you get the simplified expression:
[tex]\[
5x^4 - 37x^3 - 6x^2 + 41x - 6
\][/tex]
Thus, the correct answer is D: [tex]\(5x^4 - 37x^3 - 6x^2 + 41x - 6\)[/tex].
The original 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 Like Terms from the First Two Polynomials
First, we'll combine the terms in the first two polynomials:
- [tex]\( x^4 \)[/tex] terms: [tex]\( 5x^4 - 8x^4 = -3x^4 \)[/tex]
- [tex]\( x^3 \)[/tex] terms: [tex]\( -9x^3 \)[/tex] (no additional [tex]\( x^3 \)[/tex] term in the second polynomial)
- [tex]\( x^2 \)[/tex] terms: [tex]\( 4x^2 \)[/tex] (only present in the second polynomial)
- [tex]\( x \)[/tex] terms: [tex]\( 7x - 3x = 4x \)[/tex]
- Constant terms: [tex]\( -1 + 2 = 1 \)[/tex]
Combining these gives:
[tex]\[
-3x^4 - 9x^3 + 4x^2 + 4x + 1
\][/tex]
### Step 2: Simplify the Subtraction of the Last Expression
Now, we need to handle the subtraction of the last polynomial expression:
[tex]\[
- (-4x^3 + 5x - 1)(2x - 7)
\][/tex]
First, expand the expression [tex]\((-4x^3 + 5x - 1)(2x - 7)\)[/tex]:
- Multiplying [tex]\(-4x^3\)[/tex] with:
- [tex]\(2x\)[/tex] gives [tex]\(-8x^4\)[/tex]
- [tex]\(-7\)[/tex] gives [tex]\(28x^3\)[/tex]
- Multiplying [tex]\(5x\)[/tex] with:
- [tex]\(2x\)[/tex] gives [tex]\(10x^2\)[/tex]
- [tex]\(-7\)[/tex] gives [tex]\(-35x\)[/tex]
- Multiplying [tex]\(-1\)[/tex] with:
- [tex]\(2x\)[/tex] gives [tex]\(-2x\)[/tex]
- [tex]\(-7\)[/tex] gives [tex]\(7\)[/tex]
Combine the results of these products:
[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: Combine All Expressions
Now, substitute back and combine with the first part:
[tex]\[
(-3x^4 - 9x^3 + 4x^2 + 4x + 1) - (-8x^4 + 28x^3 + 10x^2 - 37x + 7)
\][/tex]
Subtract each corresponding term:
- [tex]\( x^4 \)[/tex] terms: [tex]\(-3x^4 - (-8x^4) = 5x^4\)[/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]\(4x - (-37x) = 41x\)[/tex]
- Constant terms: [tex]\(1 - 7 = -6\)[/tex]
Bringing it all together, you get the simplified expression:
[tex]\[
5x^4 - 37x^3 - 6x^2 + 41x - 6
\][/tex]
Thus, the correct answer is D: [tex]\(5x^4 - 37x^3 - 6x^2 + 41x - 6\)[/tex].
