Answer :
To find the quotient of the division of [tex]\((x^4 + 5x^3 - 3x - 15)\)[/tex] by [tex]\((x^3 - 3)\)[/tex], we need to perform polynomial long division. Let's solve this step-by-step.
1. Set up the division:
[tex]\[(x^4 + 5x^3 - 3x - 15) \div (x^3 - 3)\][/tex]
2. Divide the leading term of the dividend by the leading term of the divisor:
[tex]\[\frac{x^4}{x^3} = x\][/tex]
This is the first term of the quotient.
3. Multiply the entire divisor by this first term:
[tex]\[(x^3 - 3) \times x = x^4 - 3x\][/tex]
4. Subtract this result from the original dividend:
[tex]\[(x^4 + 5x^3 - 3x - 15) - (x^4 - 3x) = 5x^3 + 0x^2 - 3x + 15\][/tex]
5. Repeat the process with the new polynomial:
- Leading term of the new polynomial: [tex]\(5x^3\)[/tex]
- Divide by the leading term of the divisor: [tex]\(\frac{5x^3}{x^3} = 5\)[/tex]
- Multiply the divisor by this result: [tex]\[(x^3 - 3) \times 5 = 5x^3 - 15\][/tex]
- Subtract this from the current polynomial:
[tex]\[(5x^3 + 0x^2 - 3x - 15) - (5x^3 - 15) = 0x^2 - 3x\][/tex]
6. Continue until the remainder has a degree less than the divisor:
Now, [tex]\(0x^2 - 3x\)[/tex] doesn't have a degree higher than the divisor. We've obtained all parts of the quotient, and the remainder is [tex]\(-3x\)[/tex].
The quotient of [tex]\((x^4 + 5x^3 - 3x - 15)\)[/tex] divided by [tex]\((x^3 - 3)\)[/tex] is:
[tex]\[x + 5\][/tex]
And the remainder is [tex]\(-3x\)[/tex], but since the question asks only for the polynomial quotient, the answer is:
[tex]\[x + 5\][/tex]
Thus, the correct choice from the options provided is [tex]\(x + 5\)[/tex].
1. Set up the division:
[tex]\[(x^4 + 5x^3 - 3x - 15) \div (x^3 - 3)\][/tex]
2. Divide the leading term of the dividend by the leading term of the divisor:
[tex]\[\frac{x^4}{x^3} = x\][/tex]
This is the first term of the quotient.
3. Multiply the entire divisor by this first term:
[tex]\[(x^3 - 3) \times x = x^4 - 3x\][/tex]
4. Subtract this result from the original dividend:
[tex]\[(x^4 + 5x^3 - 3x - 15) - (x^4 - 3x) = 5x^3 + 0x^2 - 3x + 15\][/tex]
5. Repeat the process with the new polynomial:
- Leading term of the new polynomial: [tex]\(5x^3\)[/tex]
- Divide by the leading term of the divisor: [tex]\(\frac{5x^3}{x^3} = 5\)[/tex]
- Multiply the divisor by this result: [tex]\[(x^3 - 3) \times 5 = 5x^3 - 15\][/tex]
- Subtract this from the current polynomial:
[tex]\[(5x^3 + 0x^2 - 3x - 15) - (5x^3 - 15) = 0x^2 - 3x\][/tex]
6. Continue until the remainder has a degree less than the divisor:
Now, [tex]\(0x^2 - 3x\)[/tex] doesn't have a degree higher than the divisor. We've obtained all parts of the quotient, and the remainder is [tex]\(-3x\)[/tex].
The quotient of [tex]\((x^4 + 5x^3 - 3x - 15)\)[/tex] divided by [tex]\((x^3 - 3)\)[/tex] is:
[tex]\[x + 5\][/tex]
And the remainder is [tex]\(-3x\)[/tex], but since the question asks only for the polynomial quotient, the answer is:
[tex]\[x + 5\][/tex]
Thus, the correct choice from the options provided is [tex]\(x + 5\)[/tex].
