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 follow these steps:
1. Set Up the Division:
We want to divide the polynomial [tex]\(x^4 + 5x^3 - 3x - 15\)[/tex] by [tex]\(x^3 - 3\)[/tex]. The result will be a polynomial quotient and possibly a remainder.
2. Perform Polynomial Division:
- Determine how many times the leading term of the divisor, [tex]\(x^3\)[/tex], can be multiplied to match the leading term of the dividend, [tex]\(x^4\)[/tex].
- Divide the dividend's leading term, [tex]\(x^4\)[/tex], by the leading term of the divisor, [tex]\(x^3\)[/tex]. This gives a quotient term of [tex]\(x\)[/tex].
3. Multiply and Subtract:
- Multiply the entire divisor [tex]\(x^3 - 3\)[/tex] by the quotient term [tex]\(x\)[/tex]:
[tex]\[
x \cdot (x^3 - 3) = x^4 - 3x
\][/tex]
- Subtract this result from the original polynomial:
[tex]\[
(x^4 + 5x^3 - 3x - 15) - (x^4 - 3x) = 5x^3 + 0x^2 - 15
\][/tex]
4. Repeat the Process:
- Now, repeat the division process with the new polynomial [tex]\(5x^3 + 0x^2 - 3x - 15\)[/tex] and [tex]\(x^3 - 3\)[/tex].
- Divide the leading term, [tex]\(5x^3\)[/tex], by [tex]\(x^3\)[/tex] to get the next term of the quotient, which is [tex]\(+5\)[/tex].
5. Multiply and Subtract Again:
- Multiply the entire divisor [tex]\(x^3 - 3\)[/tex] by [tex]\(5\)[/tex]:
[tex]\[
5 \cdot (x^3 - 3) = 5x^3 - 15
\][/tex]
- Subtract this result from the new polynomial:
[tex]\[
(5x^3 - 3x - 15) - (5x^3 - 15) = 0
\][/tex]
- The result is a remainder of [tex]\(0\)[/tex].
6. Quotient:
- Collecting all the terms together, the quotient of the division is [tex]\(x + 5\)[/tex].
Therefore, the quotient of [tex]\((x^4 + 5x^3 - 3x - 15)\)[/tex] divided by [tex]\((x^3 - 3)\)[/tex] is [tex]\((x + 5)\)[/tex].
1. Set Up the Division:
We want to divide the polynomial [tex]\(x^4 + 5x^3 - 3x - 15\)[/tex] by [tex]\(x^3 - 3\)[/tex]. The result will be a polynomial quotient and possibly a remainder.
2. Perform Polynomial Division:
- Determine how many times the leading term of the divisor, [tex]\(x^3\)[/tex], can be multiplied to match the leading term of the dividend, [tex]\(x^4\)[/tex].
- Divide the dividend's leading term, [tex]\(x^4\)[/tex], by the leading term of the divisor, [tex]\(x^3\)[/tex]. This gives a quotient term of [tex]\(x\)[/tex].
3. Multiply and Subtract:
- Multiply the entire divisor [tex]\(x^3 - 3\)[/tex] by the quotient term [tex]\(x\)[/tex]:
[tex]\[
x \cdot (x^3 - 3) = x^4 - 3x
\][/tex]
- Subtract this result from the original polynomial:
[tex]\[
(x^4 + 5x^3 - 3x - 15) - (x^4 - 3x) = 5x^3 + 0x^2 - 15
\][/tex]
4. Repeat the Process:
- Now, repeat the division process with the new polynomial [tex]\(5x^3 + 0x^2 - 3x - 15\)[/tex] and [tex]\(x^3 - 3\)[/tex].
- Divide the leading term, [tex]\(5x^3\)[/tex], by [tex]\(x^3\)[/tex] to get the next term of the quotient, which is [tex]\(+5\)[/tex].
5. Multiply and Subtract Again:
- Multiply the entire divisor [tex]\(x^3 - 3\)[/tex] by [tex]\(5\)[/tex]:
[tex]\[
5 \cdot (x^3 - 3) = 5x^3 - 15
\][/tex]
- Subtract this result from the new polynomial:
[tex]\[
(5x^3 - 3x - 15) - (5x^3 - 15) = 0
\][/tex]
- The result is a remainder of [tex]\(0\)[/tex].
6. Quotient:
- Collecting all the terms together, the quotient of the division is [tex]\(x + 5\)[/tex].
Therefore, the quotient of [tex]\((x^4 + 5x^3 - 3x - 15)\)[/tex] divided by [tex]\((x^3 - 3)\)[/tex] is [tex]\((x + 5)\)[/tex].
