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Answer :
- Expand the product: $(-4 x^3+5 x-1)(2 x-7) = -8x^4 + 28x^3 + 10x^2 - 37x + 7$.
- Add the first two polynomials: $(5 x^4-9 x^3+7 x-1)+(-8 x^4+4 x^2-3 x+2) = -3x^4 - 9x^3 + 4x^2 + 4x + 1$.
- Subtract the expanded product from the sum: $(-3x^4 - 9x^3 + 4x^2 + 4x + 1) - (-8x^4 + 28x^3 + 10x^2 - 37x + 7) = 5x^4 - 37x^3 - 6x^2 + 41x - 6$.
- The simplified polynomial is $5 x^4-37 x^3-6 x^2+41 x-6$, which corresponds to option C. $\boxed{5 x^4-37 x^3-6 x^2+41 x-6}$
### Explanation
1. Understanding the Problem
We are asked to simplify the polynomial expression $(5 x^4-9 x^3+7 x-1)+(-8 x^4+4 x^2-3 x+2)-(-4 x^3+5 x-1)(2 x-7)$. Let's break this down step by step.
2. Expanding the Product
First, we need to expand the product $(-4 x^3+5 x-1)(2 x-7)$.
$(-4 x^3+5 x-1)(2 x-7) = -4x^3(2x - 7) + 5x(2x - 7) - 1(2x - 7)$
$= -8x^4 + 28x^3 + 10x^2 - 35x - 2x + 7$
$= -8x^4 + 28x^3 + 10x^2 - 37x + 7$
3. Adding the First Two Polynomials
Now, we need to subtract this expanded product from the sum of the first two polynomials.
$(5 x^4-9 x^3+7 x-1)+(-8 x^4+4 x^2-3 x+2) = (5x^4 - 8x^4) + (-9x^3) + (4x^2) + (7x - 3x) + (-1 + 2)$
$= -3x^4 - 9x^3 + 4x^2 + 4x + 1$
4. Subtracting the Expanded Product
Next, we subtract the expanded product from the sum:
$(-3x^4 - 9x^3 + 4x^2 + 4x + 1) - (-8x^4 + 28x^3 + 10x^2 - 37x + 7)$
$= -3x^4 - 9x^3 + 4x^2 + 4x + 1 + 8x^4 - 28x^3 - 10x^2 + 37x - 7$
$= (-3x^4 + 8x^4) + (-9x^3 - 28x^3) + (4x^2 - 10x^2) + (4x + 37x) + (1 - 7)$
$= 5x^4 - 37x^3 - 6x^2 + 41x - 6$
5. Final Answer
The simplified polynomial expression is $5 x^4-37 x^3-6 x^2+41 x-6$. Comparing this with the given options, we see that it matches option C.
6. Conclusion
Therefore, the correct answer is C. $5 x^4-37 x^3-6 x^2+41 x-6$
### Examples
Polynomial simplification is used in various fields such as engineering, physics, and computer graphics. For example, when designing a bridge, engineers use polynomial equations to model the load distribution and structural integrity. Simplifying these equations helps in efficient calculations and accurate predictions of the bridge's behavior under different conditions.
- Add the first two polynomials: $(5 x^4-9 x^3+7 x-1)+(-8 x^4+4 x^2-3 x+2) = -3x^4 - 9x^3 + 4x^2 + 4x + 1$.
- Subtract the expanded product from the sum: $(-3x^4 - 9x^3 + 4x^2 + 4x + 1) - (-8x^4 + 28x^3 + 10x^2 - 37x + 7) = 5x^4 - 37x^3 - 6x^2 + 41x - 6$.
- The simplified polynomial is $5 x^4-37 x^3-6 x^2+41 x-6$, which corresponds to option C. $\boxed{5 x^4-37 x^3-6 x^2+41 x-6}$
### Explanation
1. Understanding the Problem
We are asked to simplify the polynomial expression $(5 x^4-9 x^3+7 x-1)+(-8 x^4+4 x^2-3 x+2)-(-4 x^3+5 x-1)(2 x-7)$. Let's break this down step by step.
2. Expanding the Product
First, we need to expand the product $(-4 x^3+5 x-1)(2 x-7)$.
$(-4 x^3+5 x-1)(2 x-7) = -4x^3(2x - 7) + 5x(2x - 7) - 1(2x - 7)$
$= -8x^4 + 28x^3 + 10x^2 - 35x - 2x + 7$
$= -8x^4 + 28x^3 + 10x^2 - 37x + 7$
3. Adding the First Two Polynomials
Now, we need to subtract this expanded product from the sum of the first two polynomials.
$(5 x^4-9 x^3+7 x-1)+(-8 x^4+4 x^2-3 x+2) = (5x^4 - 8x^4) + (-9x^3) + (4x^2) + (7x - 3x) + (-1 + 2)$
$= -3x^4 - 9x^3 + 4x^2 + 4x + 1$
4. Subtracting the Expanded Product
Next, we subtract the expanded product from the sum:
$(-3x^4 - 9x^3 + 4x^2 + 4x + 1) - (-8x^4 + 28x^3 + 10x^2 - 37x + 7)$
$= -3x^4 - 9x^3 + 4x^2 + 4x + 1 + 8x^4 - 28x^3 - 10x^2 + 37x - 7$
$= (-3x^4 + 8x^4) + (-9x^3 - 28x^3) + (4x^2 - 10x^2) + (4x + 37x) + (1 - 7)$
$= 5x^4 - 37x^3 - 6x^2 + 41x - 6$
5. Final Answer
The simplified polynomial expression is $5 x^4-37 x^3-6 x^2+41 x-6$. Comparing this with the given options, we see that it matches option C.
6. Conclusion
Therefore, the correct answer is C. $5 x^4-37 x^3-6 x^2+41 x-6$
### Examples
Polynomial simplification is used in various fields such as engineering, physics, and computer graphics. For example, when designing a bridge, engineers use polynomial equations to model the load distribution and structural integrity. Simplifying these equations helps in efficient calculations and accurate predictions of the bridge's behavior under different conditions.
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