Answer :
Let's find the quotient when we divide the polynomial [tex]\(x^4 + 5x^3 - 3x - 15\)[/tex] by [tex]\(x^3 - 3\)[/tex].
We'll perform polynomial long division to find the quotient.
1. Divide the leading term of the dividend by the leading term of the divisor:
[tex]\[
\frac{x^4}{x^3} = x
\][/tex]
2. Multiply the entire divisor by this result and subtract from the dividend:
- Multiply: [tex]\((x)(x^3 - 3) = x^4 - 3x\)[/tex]
- Subtract from the dividend:
[tex]\[
\begin{array}{r}
x^4 + 5x^3 - 3x - 15 \\
-(x^4 - 3x) \\
\hline
5x^3 + 0x - 15
\end{array}
\][/tex]
3. Repeat the process with the new polynomial:
- Divide the leading term of the new dividend by the leading term of the divisor:
[tex]\[
\frac{5x^3}{x^3} = 5
\][/tex]
- Multiply and subtract:
- Multiply: [tex]\((5)(x^3 - 3) = 5x^3 - 15\)[/tex]
- Subtract:
[tex]\[
\begin{array}{r}
5x^3 + 0x - 15 \\
-(5x^3 - 15) \\
\hline
0x
\end{array}
\][/tex]
Since the remainder is now zero, the division process ends here.
Therefore, the quotient is [tex]\(x + 5\)[/tex].
The answer to the given problem is:
[tex]\[ x + 5 \][/tex]
We'll perform polynomial long division to find the quotient.
1. Divide the leading term of the dividend by the leading term of the divisor:
[tex]\[
\frac{x^4}{x^3} = x
\][/tex]
2. Multiply the entire divisor by this result and subtract from the dividend:
- Multiply: [tex]\((x)(x^3 - 3) = x^4 - 3x\)[/tex]
- Subtract from the dividend:
[tex]\[
\begin{array}{r}
x^4 + 5x^3 - 3x - 15 \\
-(x^4 - 3x) \\
\hline
5x^3 + 0x - 15
\end{array}
\][/tex]
3. Repeat the process with the new polynomial:
- Divide the leading term of the new dividend by the leading term of the divisor:
[tex]\[
\frac{5x^3}{x^3} = 5
\][/tex]
- Multiply and subtract:
- Multiply: [tex]\((5)(x^3 - 3) = 5x^3 - 15\)[/tex]
- Subtract:
[tex]\[
\begin{array}{r}
5x^3 + 0x - 15 \\
-(5x^3 - 15) \\
\hline
0x
\end{array}
\][/tex]
Since the remainder is now zero, the division process ends here.
Therefore, the quotient is [tex]\(x + 5\)[/tex].
The answer to the given problem is:
[tex]\[ x + 5 \][/tex]
