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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^7+5x^6-6x^4-30x^3+9x+45[/tex]

D. [tex]x+5[/tex]

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]

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