Answer :
Sure! To simplify the given polynomial expression, follow these steps:
### Step 1: Write Down the Expression
We have three parts to consider:
[tex]\[ (5x^4 - 9x^3 + 7x - 1) + (-8x^4 + 4x^2 - 3x + 2) - (-4x^3 + 5x - 1)(2x - 7) \][/tex]
### Step 2: Simplify the Polynomial Multiplication
First, handle the multiplication in the third part:
[tex]\[ (-4x^3 + 5x - 1)(2x - 7) \][/tex]
Use the distributive property (FOIL method for binomials) to expand this:
[tex]\[ (-4x^3)(2x) + (-4x^3)(-7) + (5x)(2x) + (5x)(-7) + (-1)(2x) + (-1)(-7) \][/tex]
Now, simplify each term:
[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 3: Rewrite the Expression
Substitute the result back into the main expression:
[tex]\[ (5x^4 - 9x^3 + 7x - 1) + (-8x^4 + 4x^2 - 3x + 2) - (-8x^4 + 28x^3 + 10x^2 - 37x + 7) \][/tex]
### Step 4: Distribute the Negative Sign in the Subtraction
Distribute the negative sign to each term in the third part:
[tex]\[ = 5x^4 - 9x^3 + 7x - 1 + (-8x^4 + 4x^2 - 3x + 2) + 8x^4 - 28x^3 - 10x^2 + 37x - 7 \][/tex]
### Step 5: Combine Like Terms
Now, combine like terms from all parts:
- [tex]\( x^4 \)[/tex] terms:
[tex]\[ 5x^4 - 8x^4 + 8x^4 = 5x^4 \][/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]\[ 7x - 3x + 37x = 41x \][/tex]
- Constant terms:
[tex]\[ -1 + 2 - 7 = -6 \][/tex]
So, combining all these we get:
[tex]\[ 5x^4 - 37x^3 - 6x^2 + 41x - 6 \][/tex]
### Conclusion
The simplified expression is:
[tex]\[ 5x^4 - 37x^3 - 6x^2 + 41x - 6 \][/tex]
Thus, the correct answer is:
[tex]\[ \boxed{5x^4 - 37x^3 - 6x^2 + 41x - 6} \][/tex]
So, the correct option is:
[tex]\[ \boxed{D} \][/tex]
### Step 1: Write Down the Expression
We have three parts to consider:
[tex]\[ (5x^4 - 9x^3 + 7x - 1) + (-8x^4 + 4x^2 - 3x + 2) - (-4x^3 + 5x - 1)(2x - 7) \][/tex]
### Step 2: Simplify the Polynomial Multiplication
First, handle the multiplication in the third part:
[tex]\[ (-4x^3 + 5x - 1)(2x - 7) \][/tex]
Use the distributive property (FOIL method for binomials) to expand this:
[tex]\[ (-4x^3)(2x) + (-4x^3)(-7) + (5x)(2x) + (5x)(-7) + (-1)(2x) + (-1)(-7) \][/tex]
Now, simplify each term:
[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 3: Rewrite the Expression
Substitute the result back into the main expression:
[tex]\[ (5x^4 - 9x^3 + 7x - 1) + (-8x^4 + 4x^2 - 3x + 2) - (-8x^4 + 28x^3 + 10x^2 - 37x + 7) \][/tex]
### Step 4: Distribute the Negative Sign in the Subtraction
Distribute the negative sign to each term in the third part:
[tex]\[ = 5x^4 - 9x^3 + 7x - 1 + (-8x^4 + 4x^2 - 3x + 2) + 8x^4 - 28x^3 - 10x^2 + 37x - 7 \][/tex]
### Step 5: Combine Like Terms
Now, combine like terms from all parts:
- [tex]\( x^4 \)[/tex] terms:
[tex]\[ 5x^4 - 8x^4 + 8x^4 = 5x^4 \][/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]\[ 7x - 3x + 37x = 41x \][/tex]
- Constant terms:
[tex]\[ -1 + 2 - 7 = -6 \][/tex]
So, combining all these we get:
[tex]\[ 5x^4 - 37x^3 - 6x^2 + 41x - 6 \][/tex]
### Conclusion
The simplified expression is:
[tex]\[ 5x^4 - 37x^3 - 6x^2 + 41x - 6 \][/tex]
Thus, the correct answer is:
[tex]\[ \boxed{5x^4 - 37x^3 - 6x^2 + 41x - 6} \][/tex]
So, the correct option is:
[tex]\[ \boxed{D} \][/tex]
