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Answer :
Let's simplify the given polynomial expression step by step:
We have three expressions given in the problem:
1. [tex]\( (5x^4 - 9x^3 + 7x - 1) \)[/tex]
2. [tex]\( (-8x^4 + 4x^2 - 3x + 2) \)[/tex]
3. [tex]\( (-4x^3 + 5x - 1)(2x - 7) \)[/tex]
To simplify, we need to perform operations as follows:
### Step 1: Expand the third expression
We need to expand [tex]\( (-4x^3 + 5x - 1)(2x - 7) \)[/tex].
Using the distributive property (FOIL method for polynomials), we expand:
[tex]\[
(-4x^3)(2x) + (-4x^3)(-7) + (5x)(2x) + (5x)(-7) + (-1)(2x) + (-1)(-7)
\][/tex]
Simplifying each term, we get:
- [tex]\((-4x^3)(2x) = -8x^4\)[/tex]
- [tex]\((-4x^3)(-7) = 28x^3\)[/tex]
- [tex]\((5x)(2x) = 10x^2\)[/tex]
- [tex]\((5x)(-7) = -35x\)[/tex]
- [tex]\((-1)(2x) = -2x\)[/tex]
- [tex]\((-1)(-7) = 7\)[/tex]
Combine all these terms:
[tex]\[
-8x^4 + 28x^3 + 10x^2 - 35x - 2x + 7
\][/tex]
Combine like terms:
[tex]\[
-8x^4 + 28x^3 + 10x^2 - 37x + 7
\][/tex]
### Step 2: Combine all three expressions
Now, we use the results from Expanding in Step 1 and plug into the original expression:
[tex]\[
(5x^4 - 9x^3 + 7x - 1) + (-8x^4 + 4x^2 - 3x + 2) - (-8x^4 + 28x^3 + 10x^2 - 37x + 7)
\][/tex]
Now, distribute the subtraction:
[tex]\[
= 5x^4 - 9x^3 + 7x - 1 - 8x^4 + 4x^2 - 3x + 2 + 8x^4 - 28x^3 - 10x^2 + 37x - 7
\][/tex]
Now, we combine like terms:
- For [tex]\(x^4\)[/tex]: [tex]\(5x^4 - 8x^4 + 8x^4 = 5x^4\)[/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]\(7x - 3x + 37x = 41x\)[/tex]
- Constant terms: [tex]\(-1 + 2 - 7 = -6\)[/tex]
Thus, the simplified expression is:
[tex]\[ 5x^4 - 37x^3 - 6x^2 + 41x - 6 \][/tex]
Therefore, the answer is option A: [tex]\( 5x^4 - 37x^3 - 6x^2 + 41x - 6 \)[/tex].
We have three expressions given in the problem:
1. [tex]\( (5x^4 - 9x^3 + 7x - 1) \)[/tex]
2. [tex]\( (-8x^4 + 4x^2 - 3x + 2) \)[/tex]
3. [tex]\( (-4x^3 + 5x - 1)(2x - 7) \)[/tex]
To simplify, we need to perform operations as follows:
### Step 1: Expand the third expression
We need to expand [tex]\( (-4x^3 + 5x - 1)(2x - 7) \)[/tex].
Using the distributive property (FOIL method for polynomials), we expand:
[tex]\[
(-4x^3)(2x) + (-4x^3)(-7) + (5x)(2x) + (5x)(-7) + (-1)(2x) + (-1)(-7)
\][/tex]
Simplifying each term, we get:
- [tex]\((-4x^3)(2x) = -8x^4\)[/tex]
- [tex]\((-4x^3)(-7) = 28x^3\)[/tex]
- [tex]\((5x)(2x) = 10x^2\)[/tex]
- [tex]\((5x)(-7) = -35x\)[/tex]
- [tex]\((-1)(2x) = -2x\)[/tex]
- [tex]\((-1)(-7) = 7\)[/tex]
Combine all these terms:
[tex]\[
-8x^4 + 28x^3 + 10x^2 - 35x - 2x + 7
\][/tex]
Combine like terms:
[tex]\[
-8x^4 + 28x^3 + 10x^2 - 37x + 7
\][/tex]
### Step 2: Combine all three expressions
Now, we use the results from Expanding in Step 1 and plug into the original expression:
[tex]\[
(5x^4 - 9x^3 + 7x - 1) + (-8x^4 + 4x^2 - 3x + 2) - (-8x^4 + 28x^3 + 10x^2 - 37x + 7)
\][/tex]
Now, distribute the subtraction:
[tex]\[
= 5x^4 - 9x^3 + 7x - 1 - 8x^4 + 4x^2 - 3x + 2 + 8x^4 - 28x^3 - 10x^2 + 37x - 7
\][/tex]
Now, we combine like terms:
- For [tex]\(x^4\)[/tex]: [tex]\(5x^4 - 8x^4 + 8x^4 = 5x^4\)[/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]\(7x - 3x + 37x = 41x\)[/tex]
- Constant terms: [tex]\(-1 + 2 - 7 = -6\)[/tex]
Thus, the simplified expression is:
[tex]\[ 5x^4 - 37x^3 - 6x^2 + 41x - 6 \][/tex]
Therefore, the answer is option A: [tex]\( 5x^4 - 37x^3 - 6x^2 + 41x - 6 \)[/tex].
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