High School

9. Perform the polynomial operations.

a. [tex]\(\left(6x^3 + 9x^2 - 3x + 8\right) + \left(2x^4 - 6x - 20\right)\)[/tex]

b. [tex]\(\left(4x^5 + 3x^4 - 9x^2 + 11\right) - \left(8x^4 - 9x^2 - 16\right)\)[/tex]

c. [tex]\(5\left(3x^2 + 6x - 5\right)\left(2x^3 + 4x\right)\)[/tex]

Answer :

Let's solve these polynomial operations step by step:

### Part a: Polynomial Addition
We're given the problem:

[tex]\[
(6x^3 + 9x^2 - 3x + 8) + (2x^4 - 6x - 20)
\][/tex]

1. Arrange terms by decreasing powers of x:
- First polynomial: [tex]\(6x^3 + 9x^2 - 3x + 8\)[/tex]
- Second polynomial: [tex]\(2x^4 + 0x^3 + 0x^2 - 6x - 20\)[/tex]

2. Add the corresponding terms:
- [tex]\(2x^4\)[/tex] (only in the second polynomial)
- [tex]\(6x^3\)[/tex]
- [tex]\(9x^2\)[/tex]
- [tex]\(-3x - 6x = -9x\)[/tex]
- [tex]\(8 - 20 = -12\)[/tex]

So the result is:

[tex]\[
2x^4 + 6x^3 + 9x^2 - 9x - 12
\][/tex]

### Part b: Polynomial Subtraction
We're given the problem:

[tex]\[
(4x^5 + 3x^4 - 9x^2 + 11) - (8x^4 - 9x^2 - 16)
\][/tex]

1. Write each polynomial including all terms:
- First polynomial: [tex]\(4x^5 + 3x^4 + 0x^3 - 9x^2 + 0x + 11\)[/tex]
- Second polynomial: [tex]\(0x^5 + 8x^4 + 0x^3 - 9x^2 + 0x + 16\)[/tex]

2. Subtract the corresponding terms:
- [tex]\(4x^5 - 0x^5 = 4x^5\)[/tex]
- [tex]\(3x^4 - 8x^4 = -5x^4\)[/tex]
- [tex]\(0x^3 - 0x^3 = 0x^3\)[/tex] (Cancels out)
- [tex]\(-9x^2 + 9x^2 = 0x^2\)[/tex] (Cancels out)
- [tex]\(0x - 0x = 0x\)[/tex] (Cancels out)
- [tex]\(11 - (-16) = 11 + 16 = 27\)[/tex]

So the result is:

[tex]\[
4x^5 - 5x^4 + 27
\][/tex]

### Part c: Polynomial Multiplication
We're given the problem:

[tex]\[
5(3x^2 + 6x - 5)(2x^3 + 4x)
\][/tex]

1. Multiply the first expression by 5:
- [tex]\(5(3x^2 + 6x - 5) = 15x^2 + 30x - 25\)[/tex]

2. Multiply by [tex]\(2x^3 + 4x\)[/tex]:
- [tex]\( (15x^2 + 30x - 25) \times (2x^3 + 4x) \)[/tex]
- Distribute each term in the first polynomial across each term in the second polynomial and combine like terms.

3. Calculate:
- [tex]\(15x^2 \times 2x^3 = 30x^5\)[/tex]
- [tex]\(15x^2 \times 4x = 60x^3\)[/tex]
- [tex]\(30x \times 2x^3 = 60x^4\)[/tex]
- [tex]\(30x \times 4x = 120x^2\)[/tex]
- [tex]\(-25 \times 2x^3 = -50x^3\)[/tex]
- [tex]\(-25 \times 4x = -100x\)[/tex]

4. Combine all terms:
- [tex]\(30x^5 + 60x^4 + (60x^3 - 50x^3) + 120x^2 - 100x\)[/tex]

So, the result of the multiplication is:

[tex]\[
30x^5 + 60x^4 + 10x^3 + 120x^2 - 100x
\][/tex]

There you have the complete solution to the polynomial operations! If you have more questions, feel free to ask.

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