Answer :
Sure! Let's go through the process of finding the quotient of the polynomials [tex]\((x^4 + 5x^3 - 3x - 15)\)[/tex] divided by [tex]\((x^3 - 3)\)[/tex].
### Step-by-step Polynomial Long Division:
1. Set up the division problem:
- Dividend: [tex]\(x^4 + 5x^3 - 3x - 15\)[/tex]
- Divisor: [tex]\(x^3 - 3\)[/tex]
- Start by writing the dividend inside the division bracket and the divisor outside.
2. Divide the first term of the dividend by the first term of the divisor:
- Divide [tex]\(x^4\)[/tex] by [tex]\(x^3\)[/tex] to get [tex]\(x\)[/tex].
- This [tex]\(x\)[/tex] is the first term of our quotient.
3. Multiply the entire divisor by this term:
- Multiply [tex]\(x^3 - 3\)[/tex] by [tex]\(x\)[/tex] to get [tex]\(x^4 - 3x\)[/tex].
4. Subtract the result from the dividend:
- [tex]\((x^4 + 5x^3 - 3x - 15) - (x^4 - 3x)\)[/tex] results in [tex]\(5x^3 + 0x^2 + 0x - 15\)[/tex].
5. Bring down the next term if applicable:
- In our case, the rest of the polynomial is already set up: [tex]\(5x^3 - 15\)[/tex].
6. Repeat the process:
- Divide [tex]\(5x^3\)[/tex] by [tex]\(x^3\)[/tex] to get [tex]\(5\)[/tex].
- Multiply the entire divisor [tex]\(x^3 - 3\)[/tex] by [tex]\(5\)[/tex] to get [tex]\(5x^3 - 15\)[/tex].
- Subtract this from the current dividend to get any remainder (which should be zero if exactly divisible).
After following these steps, we find that the quotient is:
[tex]\[ x + 5 \][/tex]
Thus, the quotient of dividing [tex]\((x^4 + 5x^3 - 3x - 15)\)[/tex] by [tex]\((x^3 - 3)\)[/tex] is [tex]\((x + 5)\)[/tex].
### Step-by-step Polynomial Long Division:
1. Set up the division problem:
- Dividend: [tex]\(x^4 + 5x^3 - 3x - 15\)[/tex]
- Divisor: [tex]\(x^3 - 3\)[/tex]
- Start by writing the dividend inside the division bracket and the divisor outside.
2. Divide the first term of the dividend by the first term of the divisor:
- Divide [tex]\(x^4\)[/tex] by [tex]\(x^3\)[/tex] to get [tex]\(x\)[/tex].
- This [tex]\(x\)[/tex] is the first term of our quotient.
3. Multiply the entire divisor by this term:
- Multiply [tex]\(x^3 - 3\)[/tex] by [tex]\(x\)[/tex] to get [tex]\(x^4 - 3x\)[/tex].
4. Subtract the result from the dividend:
- [tex]\((x^4 + 5x^3 - 3x - 15) - (x^4 - 3x)\)[/tex] results in [tex]\(5x^3 + 0x^2 + 0x - 15\)[/tex].
5. Bring down the next term if applicable:
- In our case, the rest of the polynomial is already set up: [tex]\(5x^3 - 15\)[/tex].
6. Repeat the process:
- Divide [tex]\(5x^3\)[/tex] by [tex]\(x^3\)[/tex] to get [tex]\(5\)[/tex].
- Multiply the entire divisor [tex]\(x^3 - 3\)[/tex] by [tex]\(5\)[/tex] to get [tex]\(5x^3 - 15\)[/tex].
- Subtract this from the current dividend to get any remainder (which should be zero if exactly divisible).
After following these steps, we find that the quotient is:
[tex]\[ x + 5 \][/tex]
Thus, the quotient of dividing [tex]\((x^4 + 5x^3 - 3x - 15)\)[/tex] by [tex]\((x^3 - 3)\)[/tex] is [tex]\((x + 5)\)[/tex].
