Answer :
We want 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 step by step.
1. First Step:
Divide the leading term of the numerator, [tex]$x^4$[/tex], by the leading term of the denominator, [tex]$x^3$[/tex]. This gives:
[tex]$$
\frac{x^4}{x^3} = x.
$$[/tex]
Multiply the entire divisor by [tex]$x$[/tex]:
[tex]$$
x \cdot (x^3 - 3) = x^4 - 3x.
$$[/tex]
Now subtract this product from the original numerator:
[tex]$$
\begin{align*}
(x^4 + 5x^3 - 3x - 15) - (x^4 - 3x) &= x^4 - x^4 + 5x^3 - 3x + 3x - 15 \\
&= 5x^3 - 15.
\end{align*}
$$[/tex]
2. Second Step:
Now, divide the leading term of the new polynomial [tex]$5x^3$[/tex] by the leading term [tex]$x^3$[/tex]:
[tex]$$
\frac{5x^3}{x^3} = 5.
$$[/tex]
Multiply the divisor by [tex]$5$[/tex]:
[tex]$$
5 \cdot (x^3 - 3) = 5x^3 - 15.
$$[/tex]
Subtract this from the current remainder:
[tex]$$
(5x^3 - 15) - (5x^3 - 15) = 0.
$$[/tex]
Since there is no remainder, the division is exact.
Thus, the quotient is
[tex]$$
x + 5.
$$[/tex]
[tex]$$
x^4 + 5x^3 - 3x - 15
$$[/tex]
by
[tex]$$
x^3 - 3.
$$[/tex]
We can perform polynomial long division step by step.
1. First Step:
Divide the leading term of the numerator, [tex]$x^4$[/tex], by the leading term of the denominator, [tex]$x^3$[/tex]. This gives:
[tex]$$
\frac{x^4}{x^3} = x.
$$[/tex]
Multiply the entire divisor by [tex]$x$[/tex]:
[tex]$$
x \cdot (x^3 - 3) = x^4 - 3x.
$$[/tex]
Now subtract this product from the original numerator:
[tex]$$
\begin{align*}
(x^4 + 5x^3 - 3x - 15) - (x^4 - 3x) &= x^4 - x^4 + 5x^3 - 3x + 3x - 15 \\
&= 5x^3 - 15.
\end{align*}
$$[/tex]
2. Second Step:
Now, divide the leading term of the new polynomial [tex]$5x^3$[/tex] by the leading term [tex]$x^3$[/tex]:
[tex]$$
\frac{5x^3}{x^3} = 5.
$$[/tex]
Multiply the divisor by [tex]$5$[/tex]:
[tex]$$
5 \cdot (x^3 - 3) = 5x^3 - 15.
$$[/tex]
Subtract this from the current remainder:
[tex]$$
(5x^3 - 15) - (5x^3 - 15) = 0.
$$[/tex]
Since there is no remainder, the division is exact.
Thus, the quotient is
[tex]$$
x + 5.
$$[/tex]
