Answer :
Sure, let's solve each problem step by step.
### Problem 44: Add [tex]\(2x^3 - 5x^2 + 10x - 7\)[/tex] and [tex]\(4x^2 - 7x - 2\)[/tex]
1. Write down the polynomials:
[tex]\[(2x^3 - 5x^2 + 10x - 7)\][/tex]
[tex]\[(4x^2 - 7x - 2)\][/tex]
2. Align the polynomials by their like terms:
[tex]\[2x^3 - 5x^2 + 10x - 7\][/tex]
[tex]\[0x^3 + 4x^2 - 7x - 2\][/tex]
3. Add the coefficients of like terms:
- [tex]\(x^3\)[/tex] term: [tex]\(2\)[/tex]
- [tex]\(x^2\)[/tex] term: [tex]\(-5 + 4 = -1\)[/tex]
- [tex]\(x\)[/tex] term: [tex]\(10 - 7 = 3\)[/tex]
- Constant term: [tex]\(-7 - 2 = -9\)[/tex]
After adding the polynomials, we get:
[tex]\[2x^3 - x^2 + 3x - 9\][/tex]
So, the result is:
[tex]\[\boxed{2x^3 - x^2 + 3x - 9}\][/tex]
### Problem 45: Subtract [tex]\(9x^4 - 11x^2 + 16\)[/tex] from [tex]\(6x^4 - 20x^2\)[/tex]
1. Write down the polynomials:
[tex]\[(6x^4 - 20x^2)\][/tex]
[tex]\[(9x^4 - 11x^2 + 16)\][/tex]
2. Align the polynomials by their like terms:
[tex]\[6x^4 + 0x^3 - 20x^2 + 0x + 0\][/tex]
[tex]\[9x^4 + 0x^3 - 11x^2 + 0x + 16\][/tex]
3. Subtract the coefficients of like terms:
- [tex]\(x^4\)[/tex] term: [tex]\(6 - 9 = -3\)[/tex]
- [tex]\(x^2\)[/tex] term: [tex]\(-20 - (-11) = -20 + 11 = -9\)[/tex]
- Constant term: [tex]\(0 - 16 = -16\)[/tex] (since the second polynomial has a constant term but the first does not)
After performing the subtraction, we get:
[tex]\[-3x^4 - 9x^2 - 16\][/tex]
So, the result is:
[tex]\[\boxed{-3x^4 - 9x^2 - 16}\][/tex]
These are the final results for the given problems.
### Problem 44: Add [tex]\(2x^3 - 5x^2 + 10x - 7\)[/tex] and [tex]\(4x^2 - 7x - 2\)[/tex]
1. Write down the polynomials:
[tex]\[(2x^3 - 5x^2 + 10x - 7)\][/tex]
[tex]\[(4x^2 - 7x - 2)\][/tex]
2. Align the polynomials by their like terms:
[tex]\[2x^3 - 5x^2 + 10x - 7\][/tex]
[tex]\[0x^3 + 4x^2 - 7x - 2\][/tex]
3. Add the coefficients of like terms:
- [tex]\(x^3\)[/tex] term: [tex]\(2\)[/tex]
- [tex]\(x^2\)[/tex] term: [tex]\(-5 + 4 = -1\)[/tex]
- [tex]\(x\)[/tex] term: [tex]\(10 - 7 = 3\)[/tex]
- Constant term: [tex]\(-7 - 2 = -9\)[/tex]
After adding the polynomials, we get:
[tex]\[2x^3 - x^2 + 3x - 9\][/tex]
So, the result is:
[tex]\[\boxed{2x^3 - x^2 + 3x - 9}\][/tex]
### Problem 45: Subtract [tex]\(9x^4 - 11x^2 + 16\)[/tex] from [tex]\(6x^4 - 20x^2\)[/tex]
1. Write down the polynomials:
[tex]\[(6x^4 - 20x^2)\][/tex]
[tex]\[(9x^4 - 11x^2 + 16)\][/tex]
2. Align the polynomials by their like terms:
[tex]\[6x^4 + 0x^3 - 20x^2 + 0x + 0\][/tex]
[tex]\[9x^4 + 0x^3 - 11x^2 + 0x + 16\][/tex]
3. Subtract the coefficients of like terms:
- [tex]\(x^4\)[/tex] term: [tex]\(6 - 9 = -3\)[/tex]
- [tex]\(x^2\)[/tex] term: [tex]\(-20 - (-11) = -20 + 11 = -9\)[/tex]
- Constant term: [tex]\(0 - 16 = -16\)[/tex] (since the second polynomial has a constant term but the first does not)
After performing the subtraction, we get:
[tex]\[-3x^4 - 9x^2 - 16\][/tex]
So, the result is:
[tex]\[\boxed{-3x^4 - 9x^2 - 16}\][/tex]
These are the final results for the given problems.
