Answer :
We want to simplify the expression
[tex]$$
\left(5x^4 - 9x^3 + 7x - 1\right) + \left(-8x^4 + 4x^2 - 3x + 2\right) - \left(-4x^3 + 5x - 1\right)(2x - 7).
$$[/tex]
Let's work through the steps.
1. Expand the product:
First, expand the product
[tex]$$
\left(-4x^3 + 5x - 1\right)(2x - 7).
$$[/tex]
Multiply each term in the first polynomial by each term in the second polynomial:
- [tex]$(-4x^3) \cdot (2x) = -8x^4$[/tex]
- [tex]$(-4x^3) \cdot (-7) = 28x^3$[/tex]
- [tex]$(5x) \cdot (2x) = 10x^2$[/tex]
- [tex]$(5x) \cdot (-7) = -35x$[/tex]
- [tex]$(-1) \cdot (2x) = -2x$[/tex]
- [tex]$(-1) \cdot (-7) = 7$[/tex]
Now, combine like terms from the product:
- There is only one [tex]$x^4$[/tex] term: [tex]$-8x^4$[/tex].
- The [tex]$x^3$[/tex] term is [tex]$28x^3$[/tex].
- For [tex]$x^2$[/tex], we have [tex]$10x^2$[/tex].
- For [tex]$x$[/tex], combine [tex]$-35x$[/tex] and [tex]$-2x$[/tex] to get [tex]$-37x$[/tex].
- The constant term is [tex]$7$[/tex].
Thus, the expanded product is:
[tex]$$
-8x^4 + 28x^3 + 10x^2 - 37x + 7.
$$[/tex]
2. Substitute back into the original expression:
The original expression becomes:
[tex]$$
\left(5x^4 - 9x^3 + 7x - 1\right) + \left(-8x^4 + 4x^2 - 3x + 2\right) - \left(-8x^4 + 28x^3 + 10x^2 - 37x + 7\right).
$$[/tex]
3. Remove the parentheses and combine like terms:
Write the expression without parentheses, taking care with 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, group like terms:
- [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]
Combining all these, the simplified expression is:
[tex]$$
5x^4 - 37x^3 - 6x^2 + 41x - 6.
$$[/tex]
4. Select the correct answer:
Comparing this result with the options provided, the answer is:
A. [tex]$\;5x^4 - 37x^3 - 6x^2 + 41x - 6.$[/tex]
Thus, the final simplified polynomial is
[tex]$$
5x^4 - 37x^3 - 6x^2 + 41x - 6.
$$[/tex]
[tex]$$
\left(5x^4 - 9x^3 + 7x - 1\right) + \left(-8x^4 + 4x^2 - 3x + 2\right) - \left(-4x^3 + 5x - 1\right)(2x - 7).
$$[/tex]
Let's work through the steps.
1. Expand the product:
First, expand the product
[tex]$$
\left(-4x^3 + 5x - 1\right)(2x - 7).
$$[/tex]
Multiply each term in the first polynomial by each term in the second polynomial:
- [tex]$(-4x^3) \cdot (2x) = -8x^4$[/tex]
- [tex]$(-4x^3) \cdot (-7) = 28x^3$[/tex]
- [tex]$(5x) \cdot (2x) = 10x^2$[/tex]
- [tex]$(5x) \cdot (-7) = -35x$[/tex]
- [tex]$(-1) \cdot (2x) = -2x$[/tex]
- [tex]$(-1) \cdot (-7) = 7$[/tex]
Now, combine like terms from the product:
- There is only one [tex]$x^4$[/tex] term: [tex]$-8x^4$[/tex].
- The [tex]$x^3$[/tex] term is [tex]$28x^3$[/tex].
- For [tex]$x^2$[/tex], we have [tex]$10x^2$[/tex].
- For [tex]$x$[/tex], combine [tex]$-35x$[/tex] and [tex]$-2x$[/tex] to get [tex]$-37x$[/tex].
- The constant term is [tex]$7$[/tex].
Thus, the expanded product is:
[tex]$$
-8x^4 + 28x^3 + 10x^2 - 37x + 7.
$$[/tex]
2. Substitute back into the original expression:
The original expression becomes:
[tex]$$
\left(5x^4 - 9x^3 + 7x - 1\right) + \left(-8x^4 + 4x^2 - 3x + 2\right) - \left(-8x^4 + 28x^3 + 10x^2 - 37x + 7\right).
$$[/tex]
3. Remove the parentheses and combine like terms:
Write the expression without parentheses, taking care with 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, group like terms:
- [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]
Combining all these, the simplified expression is:
[tex]$$
5x^4 - 37x^3 - 6x^2 + 41x - 6.
$$[/tex]
4. Select the correct answer:
Comparing this result with the options provided, the answer is:
A. [tex]$\;5x^4 - 37x^3 - 6x^2 + 41x - 6.$[/tex]
Thus, the final simplified polynomial is
[tex]$$
5x^4 - 37x^3 - 6x^2 + 41x - 6.
$$[/tex]
