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. Which of the following 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 solve the problem, we need to find the quotient when dividing the polynomial [tex]\(x^4 + 5x^3 - 3x - 15\)[/tex] by [tex]\(x^3 - 3\)[/tex].

To perform the division of polynomials, we'll use polynomial long division:

1. Divide the first term of the dividend by the first term of the divisor:
[tex]\[
\frac{x^4}{x^3} = x
\][/tex]
So, the first term of our quotient is [tex]\(x\)[/tex].

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

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

4. 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]
The next term of our quotient is [tex]\(5\)[/tex].

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

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

5. Since the degree of the remainder [tex]\(0x^2 - 3x\)[/tex] is less than the degree of the divisor [tex]\(x^3 - 3\)[/tex], the division process ends here.

So, the quotient of [tex]\(x^4 + 5x^3 - 3x - 15\)[/tex] divided by [tex]\(x^3 - 3\)[/tex] is [tex]\(x + 5\)[/tex], which is one of the choices provided in the list.