Answer :
Sure, let's solve it step-by-step.
To find the quotient of the polynomial division, we need to divide [tex]\((x^4 + 5x^3 - 3x - 15)\)[/tex] by [tex]\((x^3 - 3)\)[/tex].
1. Set up the division:
[tex]\[
\frac{x^4 + 5x^3 - 3x - 15}{x^3 - 3}
\][/tex]
2. Divide the leading term of the numerator by the leading term of the denominator:
[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 and subtract from the numerator:
[tex]\[
(x^3 - 3) \cdot x = x^4 - 3x
\][/tex]
Now subtract this from the original polynomial:
[tex]\[
(x^4 + 5x^3 - 3x - 15) - (x^4 - 3x) = 5x^3 - 15
\][/tex]
4. Repeat the process with the new polynomial [tex]\(5x^3 - 15\)[/tex]:
- Divide the 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].
- Multiply the entire divisor by this term and subtract:
[tex]\[
(x^3 - 3) \cdot 5 = 5x^3 - 15
\][/tex]
Subtract this from the current polynomial:
[tex]\[
5x^3 - 15 - (5x^3 - 15) = 0
\][/tex]
Thus, the quotient of the division [tex]\( \frac{x^4 + 5x^3 - 3x - 15}{x^3 - 3} \)[/tex] is [tex]\( x + 5 \)[/tex].
After going through these steps, we find that the correct answer is:
[tex]\[
\boxed{x + 5}
\][/tex]
To find the quotient of the polynomial division, we need to divide [tex]\((x^4 + 5x^3 - 3x - 15)\)[/tex] by [tex]\((x^3 - 3)\)[/tex].
1. Set up the division:
[tex]\[
\frac{x^4 + 5x^3 - 3x - 15}{x^3 - 3}
\][/tex]
2. Divide the leading term of the numerator by the leading term of the denominator:
[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 and subtract from the numerator:
[tex]\[
(x^3 - 3) \cdot x = x^4 - 3x
\][/tex]
Now subtract this from the original polynomial:
[tex]\[
(x^4 + 5x^3 - 3x - 15) - (x^4 - 3x) = 5x^3 - 15
\][/tex]
4. Repeat the process with the new polynomial [tex]\(5x^3 - 15\)[/tex]:
- Divide the 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].
- Multiply the entire divisor by this term and subtract:
[tex]\[
(x^3 - 3) \cdot 5 = 5x^3 - 15
\][/tex]
Subtract this from the current polynomial:
[tex]\[
5x^3 - 15 - (5x^3 - 15) = 0
\][/tex]
Thus, the quotient of the division [tex]\( \frac{x^4 + 5x^3 - 3x - 15}{x^3 - 3} \)[/tex] is [tex]\( x + 5 \)[/tex].
After going through these steps, we find that the correct answer is:
[tex]\[
\boxed{x + 5}
\][/tex]
