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Answer :
To find the quotient when dividing
[tex]$$
x^4 + 5x^3 - 3x - 15
$$[/tex]
by
[tex]$$
x^3 - 3,
$$[/tex]
we can perform polynomial long division.
1. First Division Step:
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]
Multiply the divisor by [tex]$x$[/tex]:
[tex]$$
x \cdot (x^3 - 3) = x^4 - 3x.
$$[/tex]
Subtract this from the original dividend:
[tex]$$
\begin{aligned}
&\quad \left( x^4 + 5x^3 - 3x - 15 \right) - \left( x^4 - 3x \right) \\
&= x^4 - x^4 + 5x^3 - 3x + 3x - 15 \\
&= 5x^3 - 15.
\end{aligned}
$$[/tex]
2. Second Division Step:
Now, divide the new leading term, [tex]$5x^3$[/tex], by [tex]$x^3$[/tex]:
[tex]$$
\frac{5x^3}{x^3} = 5.
$$[/tex]
Multiply the divisor by 5:
[tex]$$
5 \cdot (x^3 - 3) = 5x^3 - 15.
$$[/tex]
Subtract this from the remainder:
[tex]$$
\begin{aligned}
&\quad \left( 5x^3 - 15 \right) - \left(5x^3 - 15\right) \\
&= 0.
\end{aligned}
$$[/tex]
There is no remainder, and the quotient is the sum of the two terms we obtained:
[tex]$$
x + 5.
$$[/tex]
Thus, the quotient when [tex]$\displaystyle x^4+5x^3-3x-15$[/tex] is divided by [tex]$\displaystyle x^3-3$[/tex] is
[tex]$$
\boxed{x+5}.
$$[/tex]
[tex]$$
x^4 + 5x^3 - 3x - 15
$$[/tex]
by
[tex]$$
x^3 - 3,
$$[/tex]
we can perform polynomial long division.
1. First Division Step:
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]
Multiply the divisor by [tex]$x$[/tex]:
[tex]$$
x \cdot (x^3 - 3) = x^4 - 3x.
$$[/tex]
Subtract this from the original dividend:
[tex]$$
\begin{aligned}
&\quad \left( x^4 + 5x^3 - 3x - 15 \right) - \left( x^4 - 3x \right) \\
&= x^4 - x^4 + 5x^3 - 3x + 3x - 15 \\
&= 5x^3 - 15.
\end{aligned}
$$[/tex]
2. Second Division Step:
Now, divide the new leading term, [tex]$5x^3$[/tex], by [tex]$x^3$[/tex]:
[tex]$$
\frac{5x^3}{x^3} = 5.
$$[/tex]
Multiply the divisor by 5:
[tex]$$
5 \cdot (x^3 - 3) = 5x^3 - 15.
$$[/tex]
Subtract this from the remainder:
[tex]$$
\begin{aligned}
&\quad \left( 5x^3 - 15 \right) - \left(5x^3 - 15\right) \\
&= 0.
\end{aligned}
$$[/tex]
There is no remainder, and the quotient is the sum of the two terms we obtained:
[tex]$$
x + 5.
$$[/tex]
Thus, the quotient when [tex]$\displaystyle x^4+5x^3-3x-15$[/tex] is divided by [tex]$\displaystyle x^3-3$[/tex] is
[tex]$$
\boxed{x+5}.
$$[/tex]
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