Answer :
To solve the problem of finding the quotient of the polynomial [tex]\((x^4 + 5x^3 - 3x - 15)\)[/tex] divided by [tex]\((x^3 - 3)\)[/tex], we will perform polynomial long division. Here's a step-by-step explanation of the process:
1. Set Up the Division:
- Dividend: [tex]\(x^4 + 5x^3 - 3x - 15\)[/tex]
- Divisor: [tex]\(x^3 - 3\)[/tex]
2. Divide the Leading Terms:
- Divide the first term of the dividend, [tex]\(x^4\)[/tex], by the first term of the divisor, [tex]\(x^3\)[/tex].
- [tex]\(\frac{x^4}{x^3} = x\)[/tex].
3. Multiply and Subtract:
- Multiply the entire divisor, [tex]\(x^3 - 3\)[/tex], by the result, [tex]\(x\)[/tex], to get: [tex]\(x(x^3 - 3) = x^4 - 3x\)[/tex].
- Subtract [tex]\(x^4 - 3x\)[/tex] from the original dividend:
[tex]\[
(x^4 + 5x^3 - 3x - 15) - (x^4 - 3x) = 5x^3 + 0x^2 - 3x + 3x - 15
\][/tex]
- Simplify to obtain: [tex]\(5x^3 - 15\)[/tex].
4. Repeat the Process:
- Divide the new leading term, [tex]\(5x^3\)[/tex], by [tex]\(x^3\)[/tex]: [tex]\(\frac{5x^3}{x^3} = 5\)[/tex].
- Multiply and subtract again:
[tex]\[
5(x^3 - 3) = 5x^3 - 15
\][/tex]
[tex]\[
(5x^3 - 15) - (5x^3 - 15) = 0
\][/tex]
5. Reach the End:
- Since there is no remainder and we've handled all the terms, the quotient of [tex]\((x^4 + 5x^3 - 3x - 15)\)[/tex] divided by [tex]\((x^3 - 3)\)[/tex] is [tex]\(x + 5\)[/tex].
In conclusion, the quotient when dividing the given polynomials is [tex]\(x + 5\)[/tex].
1. Set Up the Division:
- Dividend: [tex]\(x^4 + 5x^3 - 3x - 15\)[/tex]
- Divisor: [tex]\(x^3 - 3\)[/tex]
2. Divide the Leading Terms:
- Divide the first term of the dividend, [tex]\(x^4\)[/tex], by the first term of the divisor, [tex]\(x^3\)[/tex].
- [tex]\(\frac{x^4}{x^3} = x\)[/tex].
3. Multiply and Subtract:
- Multiply the entire divisor, [tex]\(x^3 - 3\)[/tex], by the result, [tex]\(x\)[/tex], to get: [tex]\(x(x^3 - 3) = x^4 - 3x\)[/tex].
- Subtract [tex]\(x^4 - 3x\)[/tex] from the original dividend:
[tex]\[
(x^4 + 5x^3 - 3x - 15) - (x^4 - 3x) = 5x^3 + 0x^2 - 3x + 3x - 15
\][/tex]
- Simplify to obtain: [tex]\(5x^3 - 15\)[/tex].
4. Repeat the Process:
- Divide the new leading term, [tex]\(5x^3\)[/tex], by [tex]\(x^3\)[/tex]: [tex]\(\frac{5x^3}{x^3} = 5\)[/tex].
- Multiply and subtract again:
[tex]\[
5(x^3 - 3) = 5x^3 - 15
\][/tex]
[tex]\[
(5x^3 - 15) - (5x^3 - 15) = 0
\][/tex]
5. Reach the End:
- Since there is no remainder and we've handled all the terms, the quotient of [tex]\((x^4 + 5x^3 - 3x - 15)\)[/tex] divided by [tex]\((x^3 - 3)\)[/tex] is [tex]\(x + 5\)[/tex].
In conclusion, the quotient when dividing the given polynomials is [tex]\(x + 5\)[/tex].
