Answer :
We want to divide
[tex]$$
x^4+5x^3-3x-15
$$[/tex]
by
[tex]$$
x^3-3.
$$[/tex]
### Step 1. Arrange the Division
Write the division in the long division format:
[tex]$$
\frac{x^4+5x^3-3x-15}{x^3-3}.
$$[/tex]
### Step 2. Divide the Leading Terms
Divide the leading term of the numerator, [tex]$x^4$[/tex], by the leading term of the divisor, [tex]$x^3$[/tex]. We have:
[tex]$$
\frac{x^4}{x^3} = x.
$$[/tex]
So, the first term of the quotient is [tex]$x$[/tex].
### Step 3. Multiply and Subtract
Multiply the entire divisor by [tex]$x$[/tex]:
[tex]$$
x \cdot (x^3-3) = x^4-3x.
$$[/tex]
Subtract this product from the original polynomial:
[tex]\[
\begin{aligned}
&\quad \,\,\, \; \,\,\, x^4+5x^3-3x-15 \\
&- \, (x^4-3x) \\
&= x^4+5x^3-3x-15 - x^4+3x \\
&= 5x^3 -15.
\end{aligned}
\][/tex]
### Step 4. Continue the Division
Now, divide the new leading term [tex]$5x^3$[/tex] by [tex]$x^3$[/tex]:
[tex]$$
\frac{5x^3}{x^3} = 5.
$$[/tex]
So, the next term of the quotient is [tex]$5$[/tex].
### Step 5. Multiply and Subtract Again
Multiply the divisor by [tex]$5$[/tex]:
[tex]$$
5 \cdot (x^3-3) = 5x^3-15.
$$[/tex]
Subtract:
[tex]\[
\begin{aligned}
&\quad \,\,\, 5x^3-15 \\
&- \, (5x^3-15) \\
&= 0.
\end{aligned}
\][/tex]
Since the remainder is [tex]$0$[/tex], we have completed the division.
### Final Answer
The quotient is:
[tex]$$
x+5.
$$[/tex]
[tex]$$
x^4+5x^3-3x-15
$$[/tex]
by
[tex]$$
x^3-3.
$$[/tex]
### Step 1. Arrange the Division
Write the division in the long division format:
[tex]$$
\frac{x^4+5x^3-3x-15}{x^3-3}.
$$[/tex]
### Step 2. Divide the Leading Terms
Divide the leading term of the numerator, [tex]$x^4$[/tex], by the leading term of the divisor, [tex]$x^3$[/tex]. We have:
[tex]$$
\frac{x^4}{x^3} = x.
$$[/tex]
So, the first term of the quotient is [tex]$x$[/tex].
### Step 3. Multiply and Subtract
Multiply the entire divisor by [tex]$x$[/tex]:
[tex]$$
x \cdot (x^3-3) = x^4-3x.
$$[/tex]
Subtract this product from the original polynomial:
[tex]\[
\begin{aligned}
&\quad \,\,\, \; \,\,\, x^4+5x^3-3x-15 \\
&- \, (x^4-3x) \\
&= x^4+5x^3-3x-15 - x^4+3x \\
&= 5x^3 -15.
\end{aligned}
\][/tex]
### Step 4. Continue the Division
Now, divide the new leading term [tex]$5x^3$[/tex] by [tex]$x^3$[/tex]:
[tex]$$
\frac{5x^3}{x^3} = 5.
$$[/tex]
So, the next term of the quotient is [tex]$5$[/tex].
### Step 5. Multiply and Subtract Again
Multiply the divisor by [tex]$5$[/tex]:
[tex]$$
5 \cdot (x^3-3) = 5x^3-15.
$$[/tex]
Subtract:
[tex]\[
\begin{aligned}
&\quad \,\,\, 5x^3-15 \\
&- \, (5x^3-15) \\
&= 0.
\end{aligned}
\][/tex]
Since the remainder is [tex]$0$[/tex], we have completed the division.
### Final Answer
The quotient is:
[tex]$$
x+5.
$$[/tex]
