Answer :
We begin with the expression
[tex]$$
\left(5 x^4 - 9 x^3 + 7 x - 1\right) + \left(-8 x^4 + 4 x^2 - 3 x + 2\right) - \left(-4 x^3 + 5 x - 1\right)(2 x - 7).
$$[/tex]
Step 1. Combine the first two polynomials
Add
[tex]$$
(5x^4 - 9x^3 + 7x - 1) \quad \text{and} \quad (-8x^4 + 4x^2 - 3x + 2).
$$[/tex]
Combine like terms:
- For [tex]$x^4$[/tex]: [tex]$5x^4 + (-8x^4) = -3x^4$[/tex].
- For [tex]$x^3$[/tex]: [tex]$-9x^3$[/tex].
- For [tex]$x^2$[/tex]: [tex]$4x^2$[/tex].
- For [tex]$x$[/tex]: [tex]$7x + (-3x) = 4x$[/tex].
- For the constants: [tex]$-1 + 2 = 1$[/tex].
Thus, the sum is
[tex]$$
-3x^4 - 9x^3 + 4x^2 + 4x + 1.
$$[/tex]
Step 2. Multiply the polynomial by the factor
Multiply
[tex]$$
\left(-4 x^3 + 5 x - 1\right)(2 x - 7).
$$[/tex]
Distribute each term:
- Multiply [tex]$-4x^3$[/tex] by [tex]$2x$[/tex] and by [tex]$-7$[/tex]:
[tex]$$
-4x^3 \cdot 2x = -8x^4, \quad -4x^3 \cdot (-7) = 28x^3.
$$[/tex]
- Multiply [tex]$5x$[/tex] by [tex]$2x$[/tex] and by [tex]$-7$[/tex]:
[tex]$$
5x \cdot 2x = 10x^2, \quad 5x \cdot (-7) = -35x.
$$[/tex]
- Multiply [tex]$-1$[/tex] by [tex]$2x$[/tex] and by [tex]$-7$[/tex]:
[tex]$$
-1 \cdot 2x = -2x, \quad -1 \cdot (-7) = 7.
$$[/tex]
Now, combine like terms:
- The [tex]$x^4$[/tex] term: [tex]$-8x^4$[/tex].
- The [tex]$x^3$[/tex] term: [tex]$28x^3$[/tex].
- The [tex]$x^2$[/tex] term: [tex]$10x^2$[/tex].
- The [tex]$x$[/tex] term: [tex]$-35x - 2x = -37x$[/tex].
- The constant: [tex]$7$[/tex].
So, the product is
[tex]$$
-8x^4 + 28x^3 + 10x^2 - 37x + 7.
$$[/tex]
Step 3. Form the overall expression
Subtract the product from the sum:
[tex]$$
\left[-3x^4 - 9x^3 + 4x^2 + 4x + 1\right] - \left[-8x^4 + 28x^3 + 10x^2 - 37x + 7\right].
$$[/tex]
Distribute the subtraction sign:
[tex]$$
-3x^4 - 9x^3 + 4x^2 + 4x + 1 + 8x^4 - 28x^3 - 10x^2 + 37x - 7.
$$[/tex]
Combine like terms:
- For [tex]$x^4$[/tex]: [tex]$-3x^4 + 8x^4 = 5x^4$[/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]$4x + 37x = 41x$[/tex].
- For the constants: [tex]$1 - 7 = -6$[/tex].
Thus, the simplified expression is:
[tex]$$
5x^4 - 37x^3 - 6x^2 + 41x - 6.
$$[/tex]
This matches option D.
[tex]$$
\left(5 x^4 - 9 x^3 + 7 x - 1\right) + \left(-8 x^4 + 4 x^2 - 3 x + 2\right) - \left(-4 x^3 + 5 x - 1\right)(2 x - 7).
$$[/tex]
Step 1. Combine the first two polynomials
Add
[tex]$$
(5x^4 - 9x^3 + 7x - 1) \quad \text{and} \quad (-8x^4 + 4x^2 - 3x + 2).
$$[/tex]
Combine like terms:
- For [tex]$x^4$[/tex]: [tex]$5x^4 + (-8x^4) = -3x^4$[/tex].
- For [tex]$x^3$[/tex]: [tex]$-9x^3$[/tex].
- For [tex]$x^2$[/tex]: [tex]$4x^2$[/tex].
- For [tex]$x$[/tex]: [tex]$7x + (-3x) = 4x$[/tex].
- For the constants: [tex]$-1 + 2 = 1$[/tex].
Thus, the sum is
[tex]$$
-3x^4 - 9x^3 + 4x^2 + 4x + 1.
$$[/tex]
Step 2. Multiply the polynomial by the factor
Multiply
[tex]$$
\left(-4 x^3 + 5 x - 1\right)(2 x - 7).
$$[/tex]
Distribute each term:
- Multiply [tex]$-4x^3$[/tex] by [tex]$2x$[/tex] and by [tex]$-7$[/tex]:
[tex]$$
-4x^3 \cdot 2x = -8x^4, \quad -4x^3 \cdot (-7) = 28x^3.
$$[/tex]
- Multiply [tex]$5x$[/tex] by [tex]$2x$[/tex] and by [tex]$-7$[/tex]:
[tex]$$
5x \cdot 2x = 10x^2, \quad 5x \cdot (-7) = -35x.
$$[/tex]
- Multiply [tex]$-1$[/tex] by [tex]$2x$[/tex] and by [tex]$-7$[/tex]:
[tex]$$
-1 \cdot 2x = -2x, \quad -1 \cdot (-7) = 7.
$$[/tex]
Now, combine like terms:
- The [tex]$x^4$[/tex] term: [tex]$-8x^4$[/tex].
- The [tex]$x^3$[/tex] term: [tex]$28x^3$[/tex].
- The [tex]$x^2$[/tex] term: [tex]$10x^2$[/tex].
- The [tex]$x$[/tex] term: [tex]$-35x - 2x = -37x$[/tex].
- The constant: [tex]$7$[/tex].
So, the product is
[tex]$$
-8x^4 + 28x^3 + 10x^2 - 37x + 7.
$$[/tex]
Step 3. Form the overall expression
Subtract the product from the sum:
[tex]$$
\left[-3x^4 - 9x^3 + 4x^2 + 4x + 1\right] - \left[-8x^4 + 28x^3 + 10x^2 - 37x + 7\right].
$$[/tex]
Distribute the subtraction sign:
[tex]$$
-3x^4 - 9x^3 + 4x^2 + 4x + 1 + 8x^4 - 28x^3 - 10x^2 + 37x - 7.
$$[/tex]
Combine like terms:
- For [tex]$x^4$[/tex]: [tex]$-3x^4 + 8x^4 = 5x^4$[/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]$4x + 37x = 41x$[/tex].
- For the constants: [tex]$1 - 7 = -6$[/tex].
Thus, the simplified expression is:
[tex]$$
5x^4 - 37x^3 - 6x^2 + 41x - 6.
$$[/tex]
This matches option D.
