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Answer :
To simplify the given polynomial expression:
[tex]\[
(5x^4 - 9x^3 + 7x - 1) + (-8x^4 + 4x^2 - 3x + 2) - (-4x^3 + 5x - 1)(2x - 7)
\][/tex]
1. Distribute and Combine Like Terms:
First, distribute and expand the third part [tex]\((-4x^3 + 5x - 1)(2x - 7)\)[/tex].
Using distribution:
[tex]\[
(-4x^3 + 5x - 1)(2x - 7) = -4x^3 \cdot 2x + (-4x^3) \cdot (-7) + 5x \cdot 2x + 5x \cdot (-7) - 1 \cdot 2x - 1 \cdot (-7)
\][/tex]
This becomes:
[tex]\[
= -8x^4 + 28x^3 + 10x^2 - 35x - 2x + 7
\][/tex]
[tex]\[
= -8x^4 + 28x^3 + 10x^2 - 37x + 7
\][/tex]
2. Substitute Back & Simplify:
Now plug this back into the original expression and simplify:
[tex]\[
(5x^4 - 9x^3 + 7x - 1) + (-8x^4 + 4x^2 - 3x + 2) - (-8x^4 + 28x^3 + 10x^2 - 37x + 7)
\][/tex]
Combining like terms gives:
- For [tex]\(x^4\)[/tex]: [tex]\(5x^4 - 8x^4 + 8x^4 = 5x^4\)[/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]\(7x - 3x + 37x = 41x\)[/tex]
- For the constant: [tex]\(-1 + 2 - 7 = -6\)[/tex]
3. Write the Simplified Expression:
The simplified polynomial expression is:
[tex]\[
5x^4 - 37x^3 - 6x^2 + 41x - 6
\][/tex]
So, the answer is option B: [tex]\(5x^4 - 37x^3 - 6x^2 + 41x - 6\)[/tex].
[tex]\[
(5x^4 - 9x^3 + 7x - 1) + (-8x^4 + 4x^2 - 3x + 2) - (-4x^3 + 5x - 1)(2x - 7)
\][/tex]
1. Distribute and Combine Like Terms:
First, distribute and expand the third part [tex]\((-4x^3 + 5x - 1)(2x - 7)\)[/tex].
Using distribution:
[tex]\[
(-4x^3 + 5x - 1)(2x - 7) = -4x^3 \cdot 2x + (-4x^3) \cdot (-7) + 5x \cdot 2x + 5x \cdot (-7) - 1 \cdot 2x - 1 \cdot (-7)
\][/tex]
This becomes:
[tex]\[
= -8x^4 + 28x^3 + 10x^2 - 35x - 2x + 7
\][/tex]
[tex]\[
= -8x^4 + 28x^3 + 10x^2 - 37x + 7
\][/tex]
2. Substitute Back & Simplify:
Now plug this back into the original expression and simplify:
[tex]\[
(5x^4 - 9x^3 + 7x - 1) + (-8x^4 + 4x^2 - 3x + 2) - (-8x^4 + 28x^3 + 10x^2 - 37x + 7)
\][/tex]
Combining like terms gives:
- For [tex]\(x^4\)[/tex]: [tex]\(5x^4 - 8x^4 + 8x^4 = 5x^4\)[/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]\(7x - 3x + 37x = 41x\)[/tex]
- For the constant: [tex]\(-1 + 2 - 7 = -6\)[/tex]
3. Write the Simplified Expression:
The simplified polynomial expression is:
[tex]\[
5x^4 - 37x^3 - 6x^2 + 41x - 6
\][/tex]
So, the answer is option B: [tex]\(5x^4 - 37x^3 - 6x^2 + 41x - 6\)[/tex].
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