Answer :
Let's solve each part of the polynomial operations step by step:
### a. Polynomial Addition
We are given two polynomials to add:
1. [tex]\( 6x^3 + 9x^2 - 3x + 8 \)[/tex]
2. [tex]\( 2x^4 - 6x - 20 \)[/tex]
Step 1: Arrange the terms in descending order of their powers:
- [tex]\( 2x^4 \)[/tex] (from the second polynomial)
- [tex]\( 6x^3 \)[/tex] (from the first polynomial)
- [tex]\( 9x^2 \)[/tex] (from the first polynomial)
- [tex]\(-3x - 6x = -9x\)[/tex] (combine like terms from both polynomials)
- [tex]\(8 - 20 = -12\)[/tex] (combine constant terms)
Result:
[tex]\[ 2x^4 + 6x^3 + 9x^2 - 9x - 12 \][/tex]
### b. Polynomial Subtraction
We have the following polynomials to subtract:
1. [tex]\( 4x^5 + 3x^4 - 9x^2 + 11 \)[/tex]
2. [tex]\( 8x^4 - 9x^2 - 16 \)[/tex]
Step 1: Distribute the negative sign through the second polynomial:
- [tex]\(-8x^4 + 9x^2 + 16\)[/tex]
Step 2: Subtract the polynomials by combining like terms:
- [tex]\(4x^5\)[/tex] (only in the first polynomial)
- [tex]\(3x^4 - 8x^4 = -5x^4\)[/tex]
- [tex]\(-9x^2 + 9x^2 = 0\)[/tex] (these terms cancel each other)
- [tex]\(11 + 16 = 27\)[/tex]
Result:
[tex]\[ 4x^5 - 5x^4 + 27 \][/tex]
### c. Polynomial Multiplication
We need to multiply the following expressions:
1. [tex]\(5 \times (3x^2 + 6x - 5)\)[/tex]
2. [tex]\((2x^3 + 4x)\)[/tex]
Step 1: Multiply each term in the first polynomial by each term in the second:
- [tex]\(5 \times 3x^2 \times 2x^3 = 30x^5\)[/tex]
- [tex]\(5 \times 3x^2 \times 4x = 60x^4\)[/tex]
- [tex]\(5 \times 6x \times 2x^3 = 60x^4\)[/tex]
- [tex]\(5 \times 6x \times 4x = 120x^2\)[/tex]
- [tex]\(5 \times (-5) \times 2x^3 = -50x^3\)[/tex]
- [tex]\(5 \times (-5) \times 4x = -100x\)[/tex]
Step 2: Combine like terms:
- Combine [tex]\(60x^4 + 60x^4 = 120x^4\)[/tex]
- And combine [tex]\(-50x^3 + 10x^3 = 10x^3\)[/tex]
Result:
[tex]\[ 30x^5 + 120x^4 + 10x^3 + 120x^2 - 100x \][/tex]
These are the solutions to the given polynomial operations.
### a. Polynomial Addition
We are given two polynomials to add:
1. [tex]\( 6x^3 + 9x^2 - 3x + 8 \)[/tex]
2. [tex]\( 2x^4 - 6x - 20 \)[/tex]
Step 1: Arrange the terms in descending order of their powers:
- [tex]\( 2x^4 \)[/tex] (from the second polynomial)
- [tex]\( 6x^3 \)[/tex] (from the first polynomial)
- [tex]\( 9x^2 \)[/tex] (from the first polynomial)
- [tex]\(-3x - 6x = -9x\)[/tex] (combine like terms from both polynomials)
- [tex]\(8 - 20 = -12\)[/tex] (combine constant terms)
Result:
[tex]\[ 2x^4 + 6x^3 + 9x^2 - 9x - 12 \][/tex]
### b. Polynomial Subtraction
We have the following polynomials to subtract:
1. [tex]\( 4x^5 + 3x^4 - 9x^2 + 11 \)[/tex]
2. [tex]\( 8x^4 - 9x^2 - 16 \)[/tex]
Step 1: Distribute the negative sign through the second polynomial:
- [tex]\(-8x^4 + 9x^2 + 16\)[/tex]
Step 2: Subtract the polynomials by combining like terms:
- [tex]\(4x^5\)[/tex] (only in the first polynomial)
- [tex]\(3x^4 - 8x^4 = -5x^4\)[/tex]
- [tex]\(-9x^2 + 9x^2 = 0\)[/tex] (these terms cancel each other)
- [tex]\(11 + 16 = 27\)[/tex]
Result:
[tex]\[ 4x^5 - 5x^4 + 27 \][/tex]
### c. Polynomial Multiplication
We need to multiply the following expressions:
1. [tex]\(5 \times (3x^2 + 6x - 5)\)[/tex]
2. [tex]\((2x^3 + 4x)\)[/tex]
Step 1: Multiply each term in the first polynomial by each term in the second:
- [tex]\(5 \times 3x^2 \times 2x^3 = 30x^5\)[/tex]
- [tex]\(5 \times 3x^2 \times 4x = 60x^4\)[/tex]
- [tex]\(5 \times 6x \times 2x^3 = 60x^4\)[/tex]
- [tex]\(5 \times 6x \times 4x = 120x^2\)[/tex]
- [tex]\(5 \times (-5) \times 2x^3 = -50x^3\)[/tex]
- [tex]\(5 \times (-5) \times 4x = -100x\)[/tex]
Step 2: Combine like terms:
- Combine [tex]\(60x^4 + 60x^4 = 120x^4\)[/tex]
- And combine [tex]\(-50x^3 + 10x^3 = 10x^3\)[/tex]
Result:
[tex]\[ 30x^5 + 120x^4 + 10x^3 + 120x^2 - 100x \][/tex]
These are the solutions to the given polynomial operations.
