Answer :
Sure! Let's solve the division of the polynomial [tex]\( x^4 + 5x^3 - 3x - 15 \)[/tex] by the polynomial [tex]\( x^3 - 3 \)[/tex] using polynomial long division.
### Step-by-Step Solution:
1. Set up the division: Write down the dividend [tex]\( x^4 + 5x^3 - 3x - 15 \)[/tex] and the divisor [tex]\( x^3 - 3 \)[/tex].
2. Divide the leading terms: Divide the leading term of the dividend [tex]\( x^4 \)[/tex] by the leading term of the divisor [tex]\( x^3 \)[/tex]:
[tex]\[
\frac{x^4}{x^3} = x
\][/tex]
This gives us the first term of the quotient, which is [tex]\( x \)[/tex].
3. Multiply and subtract: Multiply the entire divisor [tex]\( x^3 - 3 \)[/tex] by the quotient term [tex]\( x \)[/tex] and subtract the result from the current dividend:
[tex]\[
\begin{align*}
& (x^4 + 5x^3 - 3x - 15) - (x \cdot (x^3 - 3)) \\
& = (x^4 + 5x^3 - 3x - 15) - (x^4 - 3x) \\
& = 5x^3 + 0x^2 - 3x - 15 + 3x \\
& = 5x^3 + 0x^2 + 0x - 15
\end{align*}
\][/tex]
4. Repeat the process:
- Divide the new leading term [tex]\( 5x^3 \)[/tex] by [tex]\( x^3 \)[/tex]:
[tex]\[
\frac{5x^3}{x^3} = 5
\][/tex]
- Multiply and subtract the entire divisor with this new term:
[tex]\[
\begin{align*}
& (5x^3 + 0x^2 + 0x - 15) - (5 \cdot (x^3 - 3)) \\
& = (5x^3 + 0x^2 + 0x - 15) - (5x^3 - 15) \\
& = 0x^3 + 0x^2 + 0x + 0
\end{align*}
\][/tex]
At this point, there are no more terms to bring down, and there is no remainder, indicating the division is complete. The quotient 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].
### Step-by-Step Solution:
1. Set up the division: Write down the dividend [tex]\( x^4 + 5x^3 - 3x - 15 \)[/tex] and the divisor [tex]\( x^3 - 3 \)[/tex].
2. Divide the leading terms: Divide the leading term of the dividend [tex]\( x^4 \)[/tex] by the leading term of the divisor [tex]\( x^3 \)[/tex]:
[tex]\[
\frac{x^4}{x^3} = x
\][/tex]
This gives us the first term of the quotient, which is [tex]\( x \)[/tex].
3. Multiply and subtract: Multiply the entire divisor [tex]\( x^3 - 3 \)[/tex] by the quotient term [tex]\( x \)[/tex] and subtract the result from the current dividend:
[tex]\[
\begin{align*}
& (x^4 + 5x^3 - 3x - 15) - (x \cdot (x^3 - 3)) \\
& = (x^4 + 5x^3 - 3x - 15) - (x^4 - 3x) \\
& = 5x^3 + 0x^2 - 3x - 15 + 3x \\
& = 5x^3 + 0x^2 + 0x - 15
\end{align*}
\][/tex]
4. Repeat the process:
- Divide the new leading term [tex]\( 5x^3 \)[/tex] by [tex]\( x^3 \)[/tex]:
[tex]\[
\frac{5x^3}{x^3} = 5
\][/tex]
- Multiply and subtract the entire divisor with this new term:
[tex]\[
\begin{align*}
& (5x^3 + 0x^2 + 0x - 15) - (5 \cdot (x^3 - 3)) \\
& = (5x^3 + 0x^2 + 0x - 15) - (5x^3 - 15) \\
& = 0x^3 + 0x^2 + 0x + 0
\end{align*}
\][/tex]
At this point, there are no more terms to bring down, and there is no remainder, indicating the division is complete. The quotient 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].
