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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]
We will do this step by step.
–––––––––––––––––––––––––––––––––––
Step 1. Add the first two polynomials
Begin by combining
[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]$$-9x^3 \quad (\text{no additional } x^3\text{ term})$$[/tex]
- For [tex]$x^2$[/tex]: [tex]$$0x^2 + 4x^2 = 4x^2.$$[/tex]
- For [tex]$x$[/tex]: [tex]$$7x + (-3x) = 4x.$$[/tex]
- Constant: [tex]$$-1 + 2 = 1.$$[/tex]
Thus, the sum is
[tex]$$
-3x^4 - 9x^3 + 4x^2 + 4x + 1.
$$[/tex]
–––––––––––––––––––––––––––––––––––
Step 2. Expand the product
Next, expand the product
[tex]$$
\left(-4x^3 + 5x - 1\right)(2x - 7).
$$[/tex]
Multiply each term in the first parenthesis by each term in the second:
1. Multiply [tex]$-4x^3$[/tex] by [tex]$2x$[/tex] and [tex]$-7$[/tex]:
- [tex]$-4x^3 \cdot 2x = -8x^4$[/tex],
- [tex]$-4x^3 \cdot (-7) = 28x^3$[/tex].
2. Multiply [tex]$5x$[/tex] by [tex]$2x$[/tex] and [tex]$-7$[/tex]:
- [tex]$5x \cdot 2x = 10x^2$[/tex],
- [tex]$5x \cdot (-7) = -35x$[/tex].
3. Multiply [tex]$-1$[/tex] by [tex]$2x$[/tex] and [tex]$-7$[/tex]:
- [tex]$-1 \cdot 2x = -2x$[/tex],
- [tex]$-1 \cdot (-7) = 7$[/tex].
Now, add the results from these multiplications:
- [tex]$x^4$[/tex] term: [tex]$$-8x^4.$$[/tex]
- [tex]$x^3$[/tex] term: [tex]$$28x^3.$$[/tex]
- [tex]$x^2$[/tex] term: [tex]$$10x^2.$$[/tex]
- [tex]$x$[/tex] terms: [tex]$$-35x - 2x = -37x.$$[/tex]
- Constant: [tex]$$7.$$[/tex]
Thus, the expanded product is
[tex]$$
-8x^4 + 28x^3 + 10x^2 - 37x + 7.
$$[/tex]
–––––––––––––––––––––––––––––––––––
Step 3. Subtract the product from the sum
Recall from Step 1 the sum:
[tex]$$
-3x^4 - 9x^3 + 4x^2 + 4x + 1,
$$[/tex]
and subtract the product obtained in Step 2:
[tex]$$
-8x^4 + 28x^3 + 10x^2 - 37x + 7.
$$[/tex]
This gives
[tex]$$
\begin{aligned}
&\bigl(-3x^4 - 9x^3 + 4x^2 + 4x + 1\bigr) - \bigl(-8x^4 + 28x^3 + 10x^2 - 37x + 7\bigr)\\[1mm]
&\quad = -3x^4 - 9x^3 + 4x^2 + 4x + 1 + 8x^4 - 28x^3 - 10x^2 + 37x - 7.
\end{aligned}
$$[/tex]
Now, 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]
- Constant: [tex]$$1 - 7 = -6.$$[/tex]
Thus, the simplified expression is
[tex]$$
5x^4 - 37x^3 - 6x^2 + 41x - 6.
$$[/tex]
–––––––––––––––––––––––––––––––––––
Conclusion
The simplified polynomial expression is
[tex]$$
5x^4 - 37x^3 - 6x^2 + 41x - 6.
$$[/tex]
Therefore, the correct answer is Option A.
[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]
We will do this step by step.
–––––––––––––––––––––––––––––––––––
Step 1. Add the first two polynomials
Begin by combining
[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]$$-9x^3 \quad (\text{no additional } x^3\text{ term})$$[/tex]
- For [tex]$x^2$[/tex]: [tex]$$0x^2 + 4x^2 = 4x^2.$$[/tex]
- For [tex]$x$[/tex]: [tex]$$7x + (-3x) = 4x.$$[/tex]
- Constant: [tex]$$-1 + 2 = 1.$$[/tex]
Thus, the sum is
[tex]$$
-3x^4 - 9x^3 + 4x^2 + 4x + 1.
$$[/tex]
–––––––––––––––––––––––––––––––––––
Step 2. Expand the product
Next, expand the product
[tex]$$
\left(-4x^3 + 5x - 1\right)(2x - 7).
$$[/tex]
Multiply each term in the first parenthesis by each term in the second:
1. Multiply [tex]$-4x^3$[/tex] by [tex]$2x$[/tex] and [tex]$-7$[/tex]:
- [tex]$-4x^3 \cdot 2x = -8x^4$[/tex],
- [tex]$-4x^3 \cdot (-7) = 28x^3$[/tex].
2. Multiply [tex]$5x$[/tex] by [tex]$2x$[/tex] and [tex]$-7$[/tex]:
- [tex]$5x \cdot 2x = 10x^2$[/tex],
- [tex]$5x \cdot (-7) = -35x$[/tex].
3. Multiply [tex]$-1$[/tex] by [tex]$2x$[/tex] and [tex]$-7$[/tex]:
- [tex]$-1 \cdot 2x = -2x$[/tex],
- [tex]$-1 \cdot (-7) = 7$[/tex].
Now, add the results from these multiplications:
- [tex]$x^4$[/tex] term: [tex]$$-8x^4.$$[/tex]
- [tex]$x^3$[/tex] term: [tex]$$28x^3.$$[/tex]
- [tex]$x^2$[/tex] term: [tex]$$10x^2.$$[/tex]
- [tex]$x$[/tex] terms: [tex]$$-35x - 2x = -37x.$$[/tex]
- Constant: [tex]$$7.$$[/tex]
Thus, the expanded product is
[tex]$$
-8x^4 + 28x^3 + 10x^2 - 37x + 7.
$$[/tex]
–––––––––––––––––––––––––––––––––––
Step 3. Subtract the product from the sum
Recall from Step 1 the sum:
[tex]$$
-3x^4 - 9x^3 + 4x^2 + 4x + 1,
$$[/tex]
and subtract the product obtained in Step 2:
[tex]$$
-8x^4 + 28x^3 + 10x^2 - 37x + 7.
$$[/tex]
This gives
[tex]$$
\begin{aligned}
&\bigl(-3x^4 - 9x^3 + 4x^2 + 4x + 1\bigr) - \bigl(-8x^4 + 28x^3 + 10x^2 - 37x + 7\bigr)\\[1mm]
&\quad = -3x^4 - 9x^3 + 4x^2 + 4x + 1 + 8x^4 - 28x^3 - 10x^2 + 37x - 7.
\end{aligned}
$$[/tex]
Now, 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]
- Constant: [tex]$$1 - 7 = -6.$$[/tex]
Thus, the simplified expression is
[tex]$$
5x^4 - 37x^3 - 6x^2 + 41x - 6.
$$[/tex]
–––––––––––––––––––––––––––––––––––
Conclusion
The simplified polynomial expression is
[tex]$$
5x^4 - 37x^3 - 6x^2 + 41x - 6.
$$[/tex]
Therefore, the correct answer is Option A.
Thanks for taking the time to read Select the correct answer Simplify the following polynomial expression tex left 5x 4 9x 3 7x 1 right left 8x 4 4x 2 3x 2. We hope the insights shared have been valuable and enhanced your understanding of the topic. Don�t hesitate to browse our website for more informative and engaging content!
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