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

Answer :

To find the quotient of the division of [tex]\(x^4 + 5x^3 - 3x - 15\)[/tex] by [tex]\(x^3 - 3\)[/tex], we can perform polynomial long division.

1. Set up the division:
Divide [tex]\(x^4 + 5x^3 - 3x - 15\)[/tex] by [tex]\(x^3 - 3\)[/tex].

2. Divide the first term of the numerator by the first term of the denominator:
[tex]\[
\frac{x^4}{x^3} = x
\][/tex]

3. Multiply the entire divisor by this result:
[tex]\[
x(x^3 - 3) = x^4 - 3x
\][/tex]

4. Subtract this from the original polynomial:
[tex]\[
(x^4 + 5x^3 - 3x - 15) - (x^4 - 3x) = 5x^3 + 0x^2 + 0x - 15
\][/tex]

5. Repeat the process with the new polynomial:
Divide the first term of the new polynomial by the first term of the divisor:
[tex]\[
\frac{5x^3}{x^3} = 5
\][/tex]

6. Multiply the entire divisor by this result:
[tex]\[
5(x^3 - 3) = 5x^3 - 15
\][/tex]

7. Subtract again:
[tex]\[
(5x^3 + 0x^2 + 0x - 15) - (5x^3 - 15) = 0
\][/tex]

No remainder is left, and the division process is complete. The final quotient is:
[tex]\[
x + 5
\][/tex]

Thus, the quotient of [tex]\((x^4+5x^3-3x-15)\)[/tex] and [tex]\((x^3-3)\)[/tex] is [tex]\(x + 5\)[/tex].

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