College

The quotient of [tex]$\left(x^4+5x^3-3x-15\right)$[/tex] and [tex]$\left(x^3-3\right)$[/tex] is a polynomial. What is the quotient?

A. [tex]$x^7+5x^6-6x^4-30x^3+9x+45$[/tex]
B. [tex]$x-5$[/tex]
C. [tex]$x+5$[/tex]
D. [tex]$x^7+5x^6+6x^4+30x^3+9x+45$[/tex]

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]