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The quotient of [tex]$x^4 + 5x^3 - 3x - 15$[/tex] and [tex]$x^3 - 3$[/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 :

Certainly! Let's solve the problem of finding the quotient of the polynomials [tex]\((x^4 + 5x^3 - 3x - 15)\)[/tex] and [tex]\((x^3 - 3)\)[/tex].

### Step-by-Step Solution:

1. Arrange the Polynomials:
- Numerator (dividend): [tex]\(x^4 + 5x^3 - 3x - 15\)[/tex]
- Denominator (divisor): [tex]\(x^3 - 3\)[/tex]

2. Perform Division:
- We want to divide the highest degree term of the numerator, [tex]\(x^4\)[/tex], by the highest degree term of the denominator, [tex]\(x^3\)[/tex].

3. First Term of the Quotient:
- Divide [tex]\(x^4\)[/tex] by [tex]\(x^3\)[/tex]:
[tex]\[
x^4 \div x^3 = x
\][/tex]
- The first term of the quotient is [tex]\(x\)[/tex].

4. Multiply and Subtract:
- Multiply the entire divisor by this term of the quotient:
[tex]\[
x \times (x^3 - 3) = x^4 - 3x
\][/tex]
- Subtract this from the original numerator:
[tex]\[
(x^4 + 5x^3 - 3x - 15) - (x^4 - 3x) = 5x^3 + 0x - 15
\][/tex]

5. Repeat the Process:
- Now, take the new polynomial [tex]\(5x^3 - 15\)[/tex].
- Divide the leading term [tex]\(5x^3\)[/tex] by [tex]\(x^3\)[/tex]:
[tex]\[
5x^3 \div x^3 = 5
\][/tex]
- Add [tex]\(5\)[/tex] to the quotient.

6. Multiply and Subtract Again:
- Multiply the divisor by this new term:
[tex]\[
5 \times (x^3 - 3) = 5x^3 - 15
\][/tex]
- Subtract:
[tex]\[
(5x^3 - 15) - (5x^3 - 15) = 0
\][/tex]

7. Conclusion:
- The remainder is zero, which indicates that [tex]\((x^4 + 5x^3 - 3x - 15)\)[/tex] divides perfectly by [tex]\((x^3 - 3)\)[/tex].

8. Final Answer:
- The quotient is:
[tex]\[
x + 5
\][/tex]

Therefore, the quotient of [tex]\((x^4 + 5x^3 - 3x - 15)\)[/tex] divided by [tex]\((x^3 - 3)\)[/tex] is the polynomial [tex]\(x + 5\)[/tex].

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