Answer :
To simplify the given polynomial expression, let's break it down step by step:
1. Write down the polynomial expression:
[tex]\[
(5x^4 - 9x^3 + 7x - 1) + (-8x^4 + 4x^2 - 3x + 2) - (-4x^3 + 5x - 1)(2x - 7)
\][/tex]
2. Simplify the first two polynomials:
Combine like terms for the expressions [tex]\((5x^4 - 9x^3 + 7x - 1)\)[/tex] and [tex]\((-8x^4 + 4x^2 - 3x + 2)\)[/tex].
[tex]\[
= 5x^4 - 9x^3 + 7x - 1 - 8x^4 + 4x^2 - 3x + 2
\][/tex]
Combine like terms:
[tex]\[
= (5x^4 - 8x^4) + (-9x^3) + 4x^2 + (7x - 3x) + (-1 + 2)
\][/tex]
[tex]\[
= -3x^4 - 9x^3 + 4x^2 + 4x + 1
\][/tex]
3. Distribute and simplify the third polynomial:
Distribute the terms over the product:
[tex]\((-4x^3 + 5x - 1)(2x - 7)\)[/tex].
[tex]\[
= (-4x^3)(2x) + (-4x^3)(-7) + (5x)(2x) + (5x)(-7) - (1)(2x) - (1)(-7)
\][/tex]
[tex]\[
= -8x^4 + 28x^3 + 10x^2 - 35x - 2x + 7
\][/tex]
Combine like terms:
[tex]\[
= -8x^4 + 28x^3 + 10x^2 - 37x + 7
\][/tex]
4. Subtract the two results:
Subtract the expression obtained in step 3 from the expression in step 2:
[tex]\[
= (-3x^4 - 9x^3 + 4x^2 + 4x + 1) - (-8x^4 + 28x^3 + 10x^2 - 37x + 7)
\][/tex]
[tex]\[
= -3x^4 + 8x^4 - 9x^3 - 28x^3 + 4x^2 - 10x^2 + 4x + 37x + 1 - 7
\][/tex]
Combine like terms:
[tex]\[
= 5x^4 - 37x^3 - 6x^2 + 41x - 6
\][/tex]
Final Answer:
The simplified polynomial expression is [tex]\(5x^4 - 37x^3 - 6x^2 + 41x - 6\)[/tex].
Therefore, the correct answer is:
B. [tex]\(5x^4 - 37x^3 - 6x^2 + 41x - 6\)[/tex]
1. Write down the polynomial expression:
[tex]\[
(5x^4 - 9x^3 + 7x - 1) + (-8x^4 + 4x^2 - 3x + 2) - (-4x^3 + 5x - 1)(2x - 7)
\][/tex]
2. Simplify the first two polynomials:
Combine like terms for the expressions [tex]\((5x^4 - 9x^3 + 7x - 1)\)[/tex] and [tex]\((-8x^4 + 4x^2 - 3x + 2)\)[/tex].
[tex]\[
= 5x^4 - 9x^3 + 7x - 1 - 8x^4 + 4x^2 - 3x + 2
\][/tex]
Combine like terms:
[tex]\[
= (5x^4 - 8x^4) + (-9x^3) + 4x^2 + (7x - 3x) + (-1 + 2)
\][/tex]
[tex]\[
= -3x^4 - 9x^3 + 4x^2 + 4x + 1
\][/tex]
3. Distribute and simplify the third polynomial:
Distribute the terms over the product:
[tex]\((-4x^3 + 5x - 1)(2x - 7)\)[/tex].
[tex]\[
= (-4x^3)(2x) + (-4x^3)(-7) + (5x)(2x) + (5x)(-7) - (1)(2x) - (1)(-7)
\][/tex]
[tex]\[
= -8x^4 + 28x^3 + 10x^2 - 35x - 2x + 7
\][/tex]
Combine like terms:
[tex]\[
= -8x^4 + 28x^3 + 10x^2 - 37x + 7
\][/tex]
4. Subtract the two results:
Subtract the expression obtained in step 3 from the expression in step 2:
[tex]\[
= (-3x^4 - 9x^3 + 4x^2 + 4x + 1) - (-8x^4 + 28x^3 + 10x^2 - 37x + 7)
\][/tex]
[tex]\[
= -3x^4 + 8x^4 - 9x^3 - 28x^3 + 4x^2 - 10x^2 + 4x + 37x + 1 - 7
\][/tex]
Combine like terms:
[tex]\[
= 5x^4 - 37x^3 - 6x^2 + 41x - 6
\][/tex]
Final Answer:
The simplified polynomial expression is [tex]\(5x^4 - 37x^3 - 6x^2 + 41x - 6\)[/tex].
Therefore, the correct answer is:
B. [tex]\(5x^4 - 37x^3 - 6x^2 + 41x - 6\)[/tex]
