Answer :
Sure! Let's tackle this problem step-by-step. We need to find the quotient of [tex]\((x^4 + 5x^3 - 3x - 15)\)[/tex] divided by [tex]\((x^3 - 3)\)[/tex].
### Step 1: Polynomial Long Division
To solve this, we'll use polynomial long division. Here's how it works:
#### Division
1. Divide the first term of the dividend by the first term of the divisor:
- [tex]\(\frac{x^4}{x^3} = x\)[/tex].
2. Multiply this result [tex]\(x\)[/tex] by the entire divisor [tex]\((x^3 - 3)\)[/tex]:
- [tex]\(x \cdot (x^3 - 3) = x^4 - 3x\)[/tex].
3. Subtract this product from the original polynomial:
- [tex]\((x^4 + 5x^3 - 3x - 15) - (x^4 - 3x) = 5x^3 + 0 - 15\)[/tex].
#### Repeat
4. Repeat the process with the new polynomial [tex]\((5x^3 - 15)\)[/tex]:
- Divide: [tex]\(\frac{5x^3}{x^3} = 5\)[/tex].
- Multiply: [tex]\(5 \times (x^3 - 3) = 5x^3 - 15\)[/tex].
- Subtract: [tex]\((5x^3 - 15) - (5x^3 - 15) = 0\)[/tex].
Now, we find that the quotient is [tex]\(x + 5\)[/tex], and there is no remainder.
### Conclusion
The quotient of [tex]\((x^4 + 5x^3 - 3x - 15)\)[/tex] divided by [tex]\((x^3 - 3)\)[/tex] is [tex]\((x + 5)\)[/tex].
So, the correct answer is [tex]\(\boxed{x + 5}\)[/tex].
### Step 1: Polynomial Long Division
To solve this, we'll use polynomial long division. Here's how it works:
#### Division
1. Divide the first term of the dividend by the first term of the divisor:
- [tex]\(\frac{x^4}{x^3} = x\)[/tex].
2. Multiply this result [tex]\(x\)[/tex] by the entire divisor [tex]\((x^3 - 3)\)[/tex]:
- [tex]\(x \cdot (x^3 - 3) = x^4 - 3x\)[/tex].
3. Subtract this product from the original polynomial:
- [tex]\((x^4 + 5x^3 - 3x - 15) - (x^4 - 3x) = 5x^3 + 0 - 15\)[/tex].
#### Repeat
4. Repeat the process with the new polynomial [tex]\((5x^3 - 15)\)[/tex]:
- Divide: [tex]\(\frac{5x^3}{x^3} = 5\)[/tex].
- Multiply: [tex]\(5 \times (x^3 - 3) = 5x^3 - 15\)[/tex].
- Subtract: [tex]\((5x^3 - 15) - (5x^3 - 15) = 0\)[/tex].
Now, we find that the quotient is [tex]\(x + 5\)[/tex], and there is no remainder.
### Conclusion
The quotient of [tex]\((x^4 + 5x^3 - 3x - 15)\)[/tex] divided by [tex]\((x^3 - 3)\)[/tex] is [tex]\((x + 5)\)[/tex].
So, the correct answer is [tex]\(\boxed{x + 5}\)[/tex].
