High School

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

Answer :

To solve the problem, we need to divide the polynomial [tex]\((x^4 + 5x^3 - 3x - 15)\)[/tex] by the polynomial [tex]\((x^3 - 3)\)[/tex] and find the quotient.

Here's how you can perform the division step-by-step:

1. Set Up the Division:
Write the dividend (the polynomial being divided) [tex]\((x^4 + 5x^3 - 3x - 15)\)[/tex] and the divisor [tex]\((x^3 - 3)\)[/tex].

2. Divide the Leading Terms:
- Divide the leading term of the dividend, [tex]\(x^4\)[/tex], by the leading term of the divisor, [tex]\(x^3\)[/tex].
- [tex]\(x^4 \div x^3 = x\)[/tex]. This is the first term of the quotient.

3. Multiply and Subtract:
- Multiply [tex]\((x^3 - 3)\)[/tex] by the first term of the quotient we found, which is [tex]\(x\)[/tex].
- This gives: [tex]\(x(x^3 - 3) = x^4 - 3x\)[/tex].
- Subtract this from the original dividend: [tex]\((x^4 + 5x^3 - 3x - 15) - (x^4 - 3x) = 5x^3 - 15\)[/tex].

4. Repeat the Process:
- Divide the new leading term, [tex]\(5x^3\)[/tex], by the leading term of the divisor, [tex]\(x^3\)[/tex].
- [tex]\(5x^3 \div x^3 = 5\)[/tex]. This is the next term of the quotient.
- Multiply and subtract again: [tex]\(5(x^3 - 3) = 5x^3 - 15\)[/tex].
- Subtract: [tex]\((5x^3 - 15) - (5x^3 - 15) = 0\)[/tex].

5. Conclusion:
Since the remainder is zero, the division is exact, and the quotient is [tex]\(x + 5\)[/tex].

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

[tex]\[
x + 5
\][/tex]