Answer :
To find the quotient of the polynomial [tex]\((x^4 + 5x^3 - 3x - 15)\)[/tex] when divided by [tex]\((x^3 - 3)\)[/tex], we need to perform polynomial long division. Here's a step-by-step guide:
1. Set up the division:
- The numerator (dividend) is [tex]\(x^4 + 5x^3 - 3x - 15\)[/tex].
- The denominator (divisor) is [tex]\(x^3 - 3\)[/tex].
2. Perform the division:
- First Division:
- Divide the leading term of the numerator, [tex]\(x^4\)[/tex], by the leading term of the denominator, [tex]\(x^3\)[/tex].
- [tex]\(\frac{x^4}{x^3} = x\)[/tex].
- Multiply the entire divisor [tex]\((x^3 - 3)\)[/tex] by [tex]\(x\)[/tex]:
[tex]\[
x \cdot (x^3 - 3) = x^4 - 3x
\][/tex]
- Subtract this from the original numerator:
[tex]\[
(x^4 + 5x^3 - 3x - 15) - (x^4 - 3x) = 5x^3 + 0x^2 - 0x - 15
\][/tex]
- Second Division:
- Divide the first term of the new polynomial, [tex]\(5x^3\)[/tex], by the first term of the divisor, [tex]\(x^3\)[/tex].
- [tex]\(\frac{5x^3}{x^3} = 5\)[/tex].
- Multiply the entire divisor by 5:
[tex]\[
5 \cdot (x^3 - 3) = 5x^3 - 15
\][/tex]
- Subtract this from the current polynomial:
[tex]\[
(5x^3 + 0x^2 - 0x - 15) - (5x^3 - 15) = 0
\][/tex]
3. Result:
- The complete expression divides evenly with no remainder. The quotient is [tex]\(x + 5\)[/tex].
Therefore, the quotient of the division of [tex]\((x^4 + 5x^3 - 3x - 15)\)[/tex] by [tex]\((x^3 - 3)\)[/tex] is [tex]\(\boxed{x + 5}\)[/tex].
1. Set up the division:
- The numerator (dividend) is [tex]\(x^4 + 5x^3 - 3x - 15\)[/tex].
- The denominator (divisor) is [tex]\(x^3 - 3\)[/tex].
2. Perform the division:
- First Division:
- Divide the leading term of the numerator, [tex]\(x^4\)[/tex], by the leading term of the denominator, [tex]\(x^3\)[/tex].
- [tex]\(\frac{x^4}{x^3} = x\)[/tex].
- Multiply the entire divisor [tex]\((x^3 - 3)\)[/tex] by [tex]\(x\)[/tex]:
[tex]\[
x \cdot (x^3 - 3) = x^4 - 3x
\][/tex]
- Subtract this from the original numerator:
[tex]\[
(x^4 + 5x^3 - 3x - 15) - (x^4 - 3x) = 5x^3 + 0x^2 - 0x - 15
\][/tex]
- Second Division:
- Divide the first term of the new polynomial, [tex]\(5x^3\)[/tex], by the first term of the divisor, [tex]\(x^3\)[/tex].
- [tex]\(\frac{5x^3}{x^3} = 5\)[/tex].
- Multiply the entire divisor by 5:
[tex]\[
5 \cdot (x^3 - 3) = 5x^3 - 15
\][/tex]
- Subtract this from the current polynomial:
[tex]\[
(5x^3 + 0x^2 - 0x - 15) - (5x^3 - 15) = 0
\][/tex]
3. Result:
- The complete expression divides evenly with no remainder. The quotient is [tex]\(x + 5\)[/tex].
Therefore, the quotient of the division of [tex]\((x^4 + 5x^3 - 3x - 15)\)[/tex] by [tex]\((x^3 - 3)\)[/tex] is [tex]\(\boxed{x + 5}\)[/tex].
