Answer :
To simplify the given polynomial expression, we need to perform addition and subtraction step by step with careful attention to the operations involved.
We start by writing down the full expression:
[tex]\[
(5x^4 - 9x^3 + 7x - 1) + (-8x^4 + 4x^2 - 3x + 2) - (-4x^3 + 5x - 1)(2x - 7)
\][/tex]
1. First: Combine the like terms from the first two polynomials:
- Collect coefficients of [tex]\(x^4\)[/tex]: [tex]\(5x^4 - 8x^4 = -3x^4\)[/tex]
- Collect coefficients of [tex]\(x^3\)[/tex]: [tex]\(-9x^3\)[/tex]
- Collect coefficients of [tex]\(x^2\)[/tex]: [tex]\(4x^2\)[/tex]
- Collect coefficients of [tex]\(x\)[/tex]: [tex]\(7x - 3x = 4x\)[/tex]
- Collect the constant terms: [tex]\(-1 + 2 = 1\)[/tex]
This simplifies the first two polynomials to:
[tex]\[
-3x^4 - 9x^3 + 4x^2 + 4x + 1
\][/tex]
2. Second: Expand the third part of the expression [tex]\((-4x^3 + 5x - 1)(2x - 7)\)[/tex]:
- Use distribution (FOIL method) to expand:
- [tex]\( -4x^3 \times 2x = -8x^4 \)[/tex]
- [tex]\( -4x^3 \times (-7) = 28x^3 \)[/tex]
- [tex]\( 5x \times 2x = 10x^2 \)[/tex]
- [tex]\( 5x \times (-7) = -35x \)[/tex]
- [tex]\( -1 \times 2x = -2x \)[/tex]
- [tex]\( -1 \times (-7) = 7 \)[/tex]
Combine these, we get:
[tex]\[
-8x^4 + 28x^3 + 10x^2 - 35x - 2x + 7 = -8x^4 + 28x^3 + 10x^2 - 37x + 7
\][/tex]
3. Third: Subtract this result from the combination of the first two polynomials:
- 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 constant term: [tex]\(1 - 7 = -6\)[/tex]
After carefully combining all the like terms, the simplified polynomial expression is:
[tex]\[
5x^4 - 37x^3 - 6x^2 + 41x - 6
\][/tex]
Therefore, the correct answer is:
D. [tex]\(5x^4 - 37x^3 - 6x^2 + 41x - 6\)[/tex]
We start by writing down the full expression:
[tex]\[
(5x^4 - 9x^3 + 7x - 1) + (-8x^4 + 4x^2 - 3x + 2) - (-4x^3 + 5x - 1)(2x - 7)
\][/tex]
1. First: Combine the like terms from the first two polynomials:
- Collect coefficients of [tex]\(x^4\)[/tex]: [tex]\(5x^4 - 8x^4 = -3x^4\)[/tex]
- Collect coefficients of [tex]\(x^3\)[/tex]: [tex]\(-9x^3\)[/tex]
- Collect coefficients of [tex]\(x^2\)[/tex]: [tex]\(4x^2\)[/tex]
- Collect coefficients of [tex]\(x\)[/tex]: [tex]\(7x - 3x = 4x\)[/tex]
- Collect the constant terms: [tex]\(-1 + 2 = 1\)[/tex]
This simplifies the first two polynomials to:
[tex]\[
-3x^4 - 9x^3 + 4x^2 + 4x + 1
\][/tex]
2. Second: Expand the third part of the expression [tex]\((-4x^3 + 5x - 1)(2x - 7)\)[/tex]:
- Use distribution (FOIL method) to expand:
- [tex]\( -4x^3 \times 2x = -8x^4 \)[/tex]
- [tex]\( -4x^3 \times (-7) = 28x^3 \)[/tex]
- [tex]\( 5x \times 2x = 10x^2 \)[/tex]
- [tex]\( 5x \times (-7) = -35x \)[/tex]
- [tex]\( -1 \times 2x = -2x \)[/tex]
- [tex]\( -1 \times (-7) = 7 \)[/tex]
Combine these, we get:
[tex]\[
-8x^4 + 28x^3 + 10x^2 - 35x - 2x + 7 = -8x^4 + 28x^3 + 10x^2 - 37x + 7
\][/tex]
3. Third: Subtract this result from the combination of the first two polynomials:
- 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 constant term: [tex]\(1 - 7 = -6\)[/tex]
After carefully combining all the like terms, the simplified polynomial expression is:
[tex]\[
5x^4 - 37x^3 - 6x^2 + 41x - 6
\][/tex]
Therefore, the correct answer is:
D. [tex]\(5x^4 - 37x^3 - 6x^2 + 41x - 6\)[/tex]
