Answer :
Sure! Let's simplify the given polynomial expression step-by-step.
The original expression is:
[tex]\((5x^2 - x^3 - 2x - 1)(-5x - 5)\)[/tex]
We will multiply each term in the first polynomial by each term in the second polynomial and then combine like terms.
### Step 1: Distribute each term
1. Multiply [tex]\(-5x\)[/tex] by each term in the first polynomial:
- [tex]\(-5x \times 5x^2 = -25x^3\)[/tex]
- [tex]\(-5x \times -x^3 = 5x^4\)[/tex]
- [tex]\(-5x \times -2x = 10x^2\)[/tex]
- [tex]\(-5x \times -1 = 5x\)[/tex]
2. Multiply [tex]\(-5\)[/tex] by each term in the first polynomial:
- [tex]\(-5 \times 5x^2 = -25x^2\)[/tex]
- [tex]\(-5 \times -x^3 = 5x^3\)[/tex]
- [tex]\(-5 \times -2x = 10x\)[/tex]
- [tex]\(-5 \times -1 = 5\)[/tex]
### Step 2: Combine all the terms
Now, let's organize and combine the like terms from the multiplication:
- For [tex]\(x^4\)[/tex] terms:
- [tex]\(5x^4\)[/tex]
- For [tex]\(x^3\)[/tex] terms:
- [tex]\(-25x^3 + 5x^3 = -20x^3\)[/tex]
- For [tex]\(x^2\)[/tex] terms:
- [tex]\(10x^2 - 25x^2 = -15x^2\)[/tex]
- For [tex]\(x\)[/tex] terms:
- [tex]\(5x + 10x = 15x\)[/tex]
- Constant term:
- [tex]\(5\)[/tex]
### Step 3: Simplified result
So, the simplified expression is:
[tex]\[ 5x^4 - 20x^3 - 15x^2 + 15x + 5 \][/tex]
Therefore, the final expression can be written and factored as:
[tex]\[ 5(x + 1)(x^3 - 5x^2 + 2x + 1) \][/tex]
This is the simplified result of the multiplication of the given polynomials.
The original expression is:
[tex]\((5x^2 - x^3 - 2x - 1)(-5x - 5)\)[/tex]
We will multiply each term in the first polynomial by each term in the second polynomial and then combine like terms.
### Step 1: Distribute each term
1. Multiply [tex]\(-5x\)[/tex] by each term in the first polynomial:
- [tex]\(-5x \times 5x^2 = -25x^3\)[/tex]
- [tex]\(-5x \times -x^3 = 5x^4\)[/tex]
- [tex]\(-5x \times -2x = 10x^2\)[/tex]
- [tex]\(-5x \times -1 = 5x\)[/tex]
2. Multiply [tex]\(-5\)[/tex] by each term in the first polynomial:
- [tex]\(-5 \times 5x^2 = -25x^2\)[/tex]
- [tex]\(-5 \times -x^3 = 5x^3\)[/tex]
- [tex]\(-5 \times -2x = 10x\)[/tex]
- [tex]\(-5 \times -1 = 5\)[/tex]
### Step 2: Combine all the terms
Now, let's organize and combine the like terms from the multiplication:
- For [tex]\(x^4\)[/tex] terms:
- [tex]\(5x^4\)[/tex]
- For [tex]\(x^3\)[/tex] terms:
- [tex]\(-25x^3 + 5x^3 = -20x^3\)[/tex]
- For [tex]\(x^2\)[/tex] terms:
- [tex]\(10x^2 - 25x^2 = -15x^2\)[/tex]
- For [tex]\(x\)[/tex] terms:
- [tex]\(5x + 10x = 15x\)[/tex]
- Constant term:
- [tex]\(5\)[/tex]
### Step 3: Simplified result
So, the simplified expression is:
[tex]\[ 5x^4 - 20x^3 - 15x^2 + 15x + 5 \][/tex]
Therefore, the final expression can be written and factored as:
[tex]\[ 5(x + 1)(x^3 - 5x^2 + 2x + 1) \][/tex]
This is the simplified result of the multiplication of the given polynomials.
