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. 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 solve the problem of finding the quotient of the polynomial division [tex]\( \frac{x^4 + 5x^3 - 3x - 15}{x^3 - 3} \)[/tex], we can perform polynomial long division. Here's a step-by-step guide:

### Step 1: Set up the division
We'll divide the polynomial [tex]\( x^4 + 5x^3 - 3x - 15 \)[/tex] (the dividend) by [tex]\( x^3 - 3 \)[/tex] (the divisor).

### Step 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 quotient term, [tex]\( x \)[/tex], will be the first term of our result.

### Step 3: Multiply and subtract
- Multiply the entire divisor [tex]\( x^3 - 3 \)[/tex] by [tex]\( x \)[/tex] (the quotient term we just found):
- [tex]\( x(x^3 - 3) = x^4 - 3x \)[/tex].

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

### Step 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].

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

- Subtract this result from the current polynomial:
- [tex]\( (5x^3 + 3x - 15) - (5x^3 - 15) = 3x \)[/tex].

### Final Result
Now, the remainder is [tex]\( 3x \)[/tex], and since it is of lower degree than the divisor, we stop here.

Thus, the quotient is [tex]\( x + 5 \)[/tex].

The quotient of the polynomial division [tex]\( \frac{x^4 + 5x^3 - 3x - 15}{x^3 - 3} \)[/tex] is [tex]\(\boxed{x + 5}\)[/tex].