Answer :
We need to divide the polynomial
[tex]$$
x^4+5x^3-3x-15
$$[/tex]
by
[tex]$$
x^3-3.
$$[/tex]
Let's perform the division step by step.
1. Write the dividend in full form (including terms with zero coefficients where necessary):
[tex]$$
x^4+5x^3+0x^2-3x-15.
$$[/tex]
2. 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]
So, the first term of the quotient is [tex]$x$[/tex].
3. Multiply the entire divisor by [tex]$x$[/tex]:
[tex]$$
x(x^3-3) = x^4-3x.
$$[/tex]
4. Subtract this product from the dividend:
[tex]$$
\begin{align*}
&\quad \left(x^4+5x^3-3x-15\right) - \left(x^4-3x\right) \\
&= x^4+5x^3-3x-15 - x^4+3x \\
&= 5x^3-15.
\end{align*}
$$[/tex]
5. Now, treat [tex]$5x^3-15$[/tex] as the new dividend. Divide its leading term, [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]$$
5(x^3-3) = 5x^3-15.
$$[/tex]
7. Subtract this from the current dividend:
[tex]$$
\begin{align*}
&\quad \left(5x^3-15\right) - \left(5x^3-15\right) \\
&= 0.
\end{align*}
$$[/tex]
Since the remainder is [tex]$0$[/tex], the division is exact and the quotient is
[tex]$$
x+5.
$$[/tex]
Thus, the quotient of the given division is [tex]$\boxed{x+5}$[/tex].
[tex]$$
x^4+5x^3-3x-15
$$[/tex]
by
[tex]$$
x^3-3.
$$[/tex]
Let's perform the division step by step.
1. Write the dividend in full form (including terms with zero coefficients where necessary):
[tex]$$
x^4+5x^3+0x^2-3x-15.
$$[/tex]
2. 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]
So, the first term of the quotient is [tex]$x$[/tex].
3. Multiply the entire divisor by [tex]$x$[/tex]:
[tex]$$
x(x^3-3) = x^4-3x.
$$[/tex]
4. Subtract this product from the dividend:
[tex]$$
\begin{align*}
&\quad \left(x^4+5x^3-3x-15\right) - \left(x^4-3x\right) \\
&= x^4+5x^3-3x-15 - x^4+3x \\
&= 5x^3-15.
\end{align*}
$$[/tex]
5. Now, treat [tex]$5x^3-15$[/tex] as the new dividend. Divide its leading term, [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]$$
5(x^3-3) = 5x^3-15.
$$[/tex]
7. Subtract this from the current dividend:
[tex]$$
\begin{align*}
&\quad \left(5x^3-15\right) - \left(5x^3-15\right) \\
&= 0.
\end{align*}
$$[/tex]
Since the remainder is [tex]$0$[/tex], the division is exact and the quotient is
[tex]$$
x+5.
$$[/tex]
Thus, the quotient of the given division is [tex]$\boxed{x+5}$[/tex].
