Answer :
To find the quotient of the polynomial division of [tex]\(x^4 + 5x^3 - 3x - 15\)[/tex] by [tex]\(x^3 - 3\)[/tex], we'll perform polynomial long division. Here’s how it works:
### Step-by-step Polynomial Long Division:
1. Arrange the Polynomials:
- Dividend: [tex]\(x^4 + 5x^3 - 3x - 15\)[/tex]
- Divisor: [tex]\(x^3 - 3\)[/tex]
2. Divide the first term:
- Divide the leading term of the dividend [tex]\(x^4\)[/tex] by the leading term of the divisor [tex]\(x^3\)[/tex]:
[tex]\[
\frac{x^4}{x^3} = x
\][/tex]
3. Multiply and Subtract:
- Multiply [tex]\(x\)[/tex] by the entire divisor [tex]\(x^3 - 3\)[/tex]:
[tex]\[
x(x^3 - 3) = x^4 - 3x
\][/tex]
- Subtract this product from the original dividend:
[tex]\[
(x^4 + 5x^3 - 3x - 15) - (x^4 - 3x) = 5x^3 + 0x - 15
\][/tex]
4. Repeat the process:
- The new dividend is now [tex]\(5x^3 - 15\)[/tex].
- Divide the leading term [tex]\(5x^3\)[/tex] by [tex]\(x^3\)[/tex]:
[tex]\[
\frac{5x^3}{x^3} = 5
\][/tex]
- Multiply [tex]\(5\)[/tex] by the entire divisor [tex]\(x^3 - 3\)[/tex]:
[tex]\[
5(x^3 - 3) = 5x^3 - 15
\][/tex]
- Subtract this from the current dividend:
[tex]\[
(5x^3 - 15) - (5x^3 - 15) = 0
\][/tex]
### Conclusion:
- The division is complete and there is no remainder.
- The quotient of the division is [tex]\(x + 5\)[/tex].
Therefore, the correct answer is: [tex]\(x + 5\)[/tex].
### Step-by-step Polynomial Long Division:
1. Arrange the Polynomials:
- Dividend: [tex]\(x^4 + 5x^3 - 3x - 15\)[/tex]
- Divisor: [tex]\(x^3 - 3\)[/tex]
2. Divide the first term:
- Divide the leading term of the dividend [tex]\(x^4\)[/tex] by the leading term of the divisor [tex]\(x^3\)[/tex]:
[tex]\[
\frac{x^4}{x^3} = x
\][/tex]
3. Multiply and Subtract:
- Multiply [tex]\(x\)[/tex] by the entire divisor [tex]\(x^3 - 3\)[/tex]:
[tex]\[
x(x^3 - 3) = x^4 - 3x
\][/tex]
- Subtract this product from the original dividend:
[tex]\[
(x^4 + 5x^3 - 3x - 15) - (x^4 - 3x) = 5x^3 + 0x - 15
\][/tex]
4. Repeat the process:
- The new dividend is now [tex]\(5x^3 - 15\)[/tex].
- Divide the leading term [tex]\(5x^3\)[/tex] by [tex]\(x^3\)[/tex]:
[tex]\[
\frac{5x^3}{x^3} = 5
\][/tex]
- Multiply [tex]\(5\)[/tex] by the entire divisor [tex]\(x^3 - 3\)[/tex]:
[tex]\[
5(x^3 - 3) = 5x^3 - 15
\][/tex]
- Subtract this from the current dividend:
[tex]\[
(5x^3 - 15) - (5x^3 - 15) = 0
\][/tex]
### Conclusion:
- The division is complete and there is no remainder.
- The quotient of the division is [tex]\(x + 5\)[/tex].
Therefore, the correct answer is: [tex]\(x + 5\)[/tex].
