Answer :
Let's simplify the given polynomial expression step by step:
We have:
[tex]\[
(5x^4 - 9x^3 + 7x - 1) + (-8x^4 + 4x^2 - 3x + 2) - ((-4x^3 + 5x - 1)(2x - 7))
\][/tex]
1. Combine the Polynomials:
Group the terms from the first two polynomials:
[tex]\[
(5x^4 - 9x^3 + 7x - 1) + (-8x^4 + 4x^2 - 3x + 2)
\][/tex]
Combine like terms:
- [tex]\(5x^4 + (-8x^4) = -3x^4\)[/tex]
- [tex]\(-9x^3\)[/tex] remains as it is
- [tex]\(7x + (-3x) = 4x\)[/tex]
- [tex]\(-1 + 2 = 1\)[/tex]
- [tex]\(4x^2\)[/tex] remains as it is
So, the expression becomes:
[tex]\[
-3x^4 - 9x^3 + 4x^2 + 4x + 1
\][/tex]
2. Distribute and Simplify the Third Polynomial:
Now, expand [tex]\((-4x^3 + 5x - 1)(2x - 7)\)[/tex].
Using the distributive property ([tex]\(a(b + c) = ab + ac\)[/tex]):
[tex]\[
(-4x^3)(2x) + (-4x^3)(-7) + (5x)(2x) + (5x)(-7) + (-1)(2x) + (-1)(-7)
\][/tex]
- [tex]\((-4x^3)(2x) = -8x^4\)[/tex]
- [tex]\((-4x^3)(-7) = 28x^3\)[/tex]
- [tex]\((5x)(2x) = 10x^2\)[/tex]
- [tex]\((5x)(-7) = -35x\)[/tex]
- [tex]\((-1)(2x) = -2x\)[/tex]
- [tex]\((-1)(-7) = 7\)[/tex]
Combine these results:
[tex]\[
-8x^4 + 28x^3 + 10x^2 - 35x - 2x + 7 = -8x^4 + 28x^3 + 10x^2 - 37x + 7
\][/tex]
3. Subtract and Simplify the Expressions:
Now, subtract the result from the combined polynomials:
[tex]\[
(-3x^4 - 9x^3 + 4x^2 + 4x + 1) - (-8x^4 + 28x^3 + 10x^2 - 37x + 7)
\][/tex]
Subtract corresponding terms (be careful with the signs):
- [tex]\(-3x^4 - (-8x^4) = 5x^4\)[/tex]
- [tex]\(-9x^3 - 28x^3 = -37x^3\)[/tex]
- [tex]\(4x^2 - 10x^2 = -6x^2\)[/tex]
- [tex]\(4x - (-37x) = 41x\)[/tex]
- [tex]\(1 - 7 = -6\)[/tex]
So, the simplified expression is:
[tex]\[
5x^4 - 37x^3 - 6x^2 + 41x - 6
\][/tex]
Therefore, the correct answer is C. [tex]\(5x^4 - 37x^3 - 6x^2 + 41x - 6\)[/tex].
We have:
[tex]\[
(5x^4 - 9x^3 + 7x - 1) + (-8x^4 + 4x^2 - 3x + 2) - ((-4x^3 + 5x - 1)(2x - 7))
\][/tex]
1. Combine the Polynomials:
Group the terms from the first two polynomials:
[tex]\[
(5x^4 - 9x^3 + 7x - 1) + (-8x^4 + 4x^2 - 3x + 2)
\][/tex]
Combine like terms:
- [tex]\(5x^4 + (-8x^4) = -3x^4\)[/tex]
- [tex]\(-9x^3\)[/tex] remains as it is
- [tex]\(7x + (-3x) = 4x\)[/tex]
- [tex]\(-1 + 2 = 1\)[/tex]
- [tex]\(4x^2\)[/tex] remains as it is
So, the expression becomes:
[tex]\[
-3x^4 - 9x^3 + 4x^2 + 4x + 1
\][/tex]
2. Distribute and Simplify the Third Polynomial:
Now, expand [tex]\((-4x^3 + 5x - 1)(2x - 7)\)[/tex].
Using the distributive property ([tex]\(a(b + c) = ab + ac\)[/tex]):
[tex]\[
(-4x^3)(2x) + (-4x^3)(-7) + (5x)(2x) + (5x)(-7) + (-1)(2x) + (-1)(-7)
\][/tex]
- [tex]\((-4x^3)(2x) = -8x^4\)[/tex]
- [tex]\((-4x^3)(-7) = 28x^3\)[/tex]
- [tex]\((5x)(2x) = 10x^2\)[/tex]
- [tex]\((5x)(-7) = -35x\)[/tex]
- [tex]\((-1)(2x) = -2x\)[/tex]
- [tex]\((-1)(-7) = 7\)[/tex]
Combine these results:
[tex]\[
-8x^4 + 28x^3 + 10x^2 - 35x - 2x + 7 = -8x^4 + 28x^3 + 10x^2 - 37x + 7
\][/tex]
3. Subtract and Simplify the Expressions:
Now, subtract the result from the combined polynomials:
[tex]\[
(-3x^4 - 9x^3 + 4x^2 + 4x + 1) - (-8x^4 + 28x^3 + 10x^2 - 37x + 7)
\][/tex]
Subtract corresponding terms (be careful with the signs):
- [tex]\(-3x^4 - (-8x^4) = 5x^4\)[/tex]
- [tex]\(-9x^3 - 28x^3 = -37x^3\)[/tex]
- [tex]\(4x^2 - 10x^2 = -6x^2\)[/tex]
- [tex]\(4x - (-37x) = 41x\)[/tex]
- [tex]\(1 - 7 = -6\)[/tex]
So, the simplified expression is:
[tex]\[
5x^4 - 37x^3 - 6x^2 + 41x - 6
\][/tex]
Therefore, the correct answer is C. [tex]\(5x^4 - 37x^3 - 6x^2 + 41x - 6\)[/tex].
