To find the quotient of the polynomials [tex]\((x^4 + 5x^3 - 3x - 15)\)[/tex] divided by [tex]\((x^3 - 3)\)[/tex], we will perform polynomial long division. Here is a step-by-step solution:
1. Set up the long division:
Place the dividend [tex]\((x^4 + 5x^3 - 3x - 15)\)[/tex] under the division bar and the divisor [tex]\((x^3 - 3)\)[/tex] outside.
2. Divide the leading term of the dividend by the leading term of the divisor:
[tex]\[
\frac{x^4}{x^3} = x
\][/tex]
So, the first term of the quotient is [tex]\(x\)[/tex].
3. Multiply the entire divisor by this term:
[tex]\[
x(x^3 - 3) = x^4 - 3x
\][/tex]
4. Subtract this product from the dividend:
[tex]\[
(x^4 + 5x^3 - 3x - 15) - (x^4 - 3x) = 5x^3 - 15
\][/tex]
5. Repeat the process with the new polynomial (5x^3 - 15):
6. Divide the new leading term by the leading term of the divisor:
[tex]\[
\frac{5x^3}{x^3} = 5
\][/tex]
So, the next term of the quotient is [tex]\(5\)[/tex].
7. Multiply the entire divisor by this term:
[tex]\[
5(x^3 - 3) = 5x^3 - 15
\][/tex]
8. Subtract this product from the new polynomial:
[tex]\[
(5x^3 - 15) - (5x^3 - 15) = 0
\][/tex]
With no remainder left after the subtraction, the quotient obtained 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]\[
\boxed{x + 5}
\][/tex]
