Answer :
To find the quotient when dividing [tex]\(x^4 + 5x^3 - 3x - 15\)[/tex] by [tex]\(x^3 - 3\)[/tex], we will perform polynomial long division.
1. Set up the division: We'll divide the polynomial [tex]\(x^4 + 5x^3 - 3x - 15\)[/tex] by [tex]\(x^3 - 3\)[/tex].
2. Divide the leading terms:
- The leading term of the dividend (the polynomial we are dividing) is [tex]\(x^4\)[/tex].
- The leading term of the divisor is [tex]\(x^3\)[/tex].
- Divide these leading terms: [tex]\(\frac{x^4}{x^3} = x\)[/tex].
- This is the first term of our quotient.
3. Multiply and Subtract:
- Multiply the entire divisor [tex]\(x^3 - 3\)[/tex] by [tex]\(x\)[/tex], giving us [tex]\(x(x^3 - 3) = x^4 - 3x\)[/tex].
- Subtract this result from the original polynomial:
[tex]\[
(x^4 + 5x^3 - 3x - 15) - (x^4 - 3x) = 5x^3 + 0x^2 + 0x - 15
\][/tex]
- Simplifying gives us [tex]\(5x^3 - 0x^2 + 0x - 15\)[/tex].
4. Repeat the process:
- Divide the leading term of the new polynomial [tex]\(5x^3\)[/tex] by the leading term of the divisor [tex]\(x^3\)[/tex].
- [tex]\(\frac{5x^3}{x^3} = 5\)[/tex].
- Add this to the quotient so far, for a quotient of [tex]\(x + 5\)[/tex].
5. Multiply and Subtract:
- Multiply the entire divisor by this new term in the quotient: [tex]\(5(x^3 - 3) = 5x^3 - 15\)[/tex].
- Subtract:
[tex]\[
(5x^3 - 15) - (5x^3 - 15) = 0
\][/tex]
- There is no remainder.
Therefore, the quotient of [tex]\((x^4 + 5x^3 - 3x - 15)\)[/tex] divided by [tex]\((x^3 - 3)\)[/tex] is [tex]\(\boxed{x + 5}\)[/tex].
1. Set up the division: We'll divide the polynomial [tex]\(x^4 + 5x^3 - 3x - 15\)[/tex] by [tex]\(x^3 - 3\)[/tex].
2. Divide the leading terms:
- The leading term of the dividend (the polynomial we are dividing) is [tex]\(x^4\)[/tex].
- The leading term of the divisor is [tex]\(x^3\)[/tex].
- Divide these leading terms: [tex]\(\frac{x^4}{x^3} = x\)[/tex].
- This is the first term of our quotient.
3. Multiply and Subtract:
- Multiply the entire divisor [tex]\(x^3 - 3\)[/tex] by [tex]\(x\)[/tex], giving us [tex]\(x(x^3 - 3) = x^4 - 3x\)[/tex].
- Subtract this result from the original polynomial:
[tex]\[
(x^4 + 5x^3 - 3x - 15) - (x^4 - 3x) = 5x^3 + 0x^2 + 0x - 15
\][/tex]
- Simplifying gives us [tex]\(5x^3 - 0x^2 + 0x - 15\)[/tex].
4. Repeat the process:
- Divide the leading term of the new polynomial [tex]\(5x^3\)[/tex] by the leading term of the divisor [tex]\(x^3\)[/tex].
- [tex]\(\frac{5x^3}{x^3} = 5\)[/tex].
- Add this to the quotient so far, for a quotient of [tex]\(x + 5\)[/tex].
5. Multiply and Subtract:
- Multiply the entire divisor by this new term in the quotient: [tex]\(5(x^3 - 3) = 5x^3 - 15\)[/tex].
- Subtract:
[tex]\[
(5x^3 - 15) - (5x^3 - 15) = 0
\][/tex]
- There is no remainder.
Therefore, the quotient of [tex]\((x^4 + 5x^3 - 3x - 15)\)[/tex] divided by [tex]\((x^3 - 3)\)[/tex] is [tex]\(\boxed{x + 5}\)[/tex].
