Answer :
We want to find the polynomial quotient when dividing
[tex]$$
x^4 + 5x^3 - 3x - 15
$$[/tex]
by
[tex]$$
x^3 - 3.
$$[/tex]
We perform polynomial long division step by step.
1. Notice that the dividend is
[tex]$$
x^4 + 5x^3 - 3x - 15,
$$[/tex]
and the divisor is
[tex]$$
x^3-3.
$$[/tex]
2. Divide the leading term of the dividend, [tex]$x^4$[/tex], by the leading term of the divisor, [tex]$x^3$[/tex]. This gives
[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 [tex]$x$[/tex]:
[tex]$$
x \cdot (x^3-3) = x^4 - 3x.
$$[/tex]
4. Subtract this product from the dividend:
[tex]$$
\begin{aligned}
\left(x^4 + 5x^3 - 3x - 15\right) - \left(x^4 - 3x\right)
&= (x^4 - x^4) + 5x^3 + (-3x + 3x) - 15 \\
&= 5x^3 - 15.
\end{aligned}
$$[/tex]
Now, the new dividend is [tex]$5x^3 - 15$[/tex].
5. Next, divide the leading term of the new dividend, [tex]$5x^3$[/tex], by the leading term of the divisor, [tex]$x^3$[/tex]:
[tex]$$
\frac{5x^3}{x^3} = 5.
$$[/tex]
So, the next term of the quotient is [tex]$5$[/tex].
6. Multiply the divisor by [tex]$5$[/tex]:
[tex]$$
5 \cdot (x^3 - 3) = 5x^3 - 15.
$$[/tex]
7. Subtract this from the current dividend:
[tex]$$
\begin{aligned}
(5x^3 - 15) - (5x^3 - 15)
&= 5x^3 - 5x^3 - 15 + 15 \\
&= 0.
\end{aligned}
$$[/tex]
The remainder is [tex]$0$[/tex].
8. Since the division process ends with a remainder of [tex]$0$[/tex], the complete quotient is the sum of the terms we obtained:
[tex]$$
x + 5.
$$[/tex]
Thus, the quotient of the division is
[tex]$$
\boxed{x+5}.
$$[/tex]
[tex]$$
x^4 + 5x^3 - 3x - 15
$$[/tex]
by
[tex]$$
x^3 - 3.
$$[/tex]
We perform polynomial long division step by step.
1. Notice that the dividend is
[tex]$$
x^4 + 5x^3 - 3x - 15,
$$[/tex]
and the divisor is
[tex]$$
x^3-3.
$$[/tex]
2. Divide the leading term of the dividend, [tex]$x^4$[/tex], by the leading term of the divisor, [tex]$x^3$[/tex]. This gives
[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 [tex]$x$[/tex]:
[tex]$$
x \cdot (x^3-3) = x^4 - 3x.
$$[/tex]
4. Subtract this product from the dividend:
[tex]$$
\begin{aligned}
\left(x^4 + 5x^3 - 3x - 15\right) - \left(x^4 - 3x\right)
&= (x^4 - x^4) + 5x^3 + (-3x + 3x) - 15 \\
&= 5x^3 - 15.
\end{aligned}
$$[/tex]
Now, the new dividend is [tex]$5x^3 - 15$[/tex].
5. Next, divide the leading term of the new dividend, [tex]$5x^3$[/tex], by the leading term of the divisor, [tex]$x^3$[/tex]:
[tex]$$
\frac{5x^3}{x^3} = 5.
$$[/tex]
So, the next term of the quotient is [tex]$5$[/tex].
6. Multiply the divisor by [tex]$5$[/tex]:
[tex]$$
5 \cdot (x^3 - 3) = 5x^3 - 15.
$$[/tex]
7. Subtract this from the current dividend:
[tex]$$
\begin{aligned}
(5x^3 - 15) - (5x^3 - 15)
&= 5x^3 - 5x^3 - 15 + 15 \\
&= 0.
\end{aligned}
$$[/tex]
The remainder is [tex]$0$[/tex].
8. Since the division process ends with a remainder of [tex]$0$[/tex], the complete quotient is the sum of the terms we obtained:
[tex]$$
x + 5.
$$[/tex]
Thus, the quotient of the division is
[tex]$$
\boxed{x+5}.
$$[/tex]
