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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 :

Sure! Let's tackle the problem step-by-step to find the quotient when dividing the polynomial [tex]\(x^4 + 5x^3 - 3x - 15\)[/tex] by [tex]\(x^3 - 3\)[/tex].

### Step 1: Set Up the Division

We will perform polynomial long division. We are dividing:

Dividend: [tex]\(x^4 + 5x^3 - 3x - 15\)[/tex]

Divisor: [tex]\(x^3 - 3\)[/tex]

### Step 2: Perform 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 the quotient is [tex]\(x\)[/tex].

2. Multiply the entire divisor by the first term of the quotient ([tex]\(x\)[/tex]) and subtract from the dividend:

[tex]\[
x(x^3 - 3) = x^4 - 3x
\][/tex]

[tex]\[
(x^4 + 5x^3 - 3x - 15) - (x^4 - 3x) = 5x^3 + 0x^2 - 0x - 15
\][/tex]

3. Repeat the process with the new polynomial:

- Divide the first term of the new dividend by the first term of the divisor:

[tex]\[
\frac{5x^3}{x^3} = 5
\][/tex]

So, the second term of the quotient is [tex]\(+5\)[/tex].

- Multiply the entire divisor by [tex]\(5\)[/tex] and subtract:

[tex]\[
5(x^3 - 3) = 5x^3 - 15
\][/tex]

[tex]\[
(5x^3 + 0x^2 - 0x - 15) - (5x^3 - 15) = 0
\][/tex]

### Step 3: Conclusion

Since subtracting results in zero, we have no remainder. Therefore, the quotient of the division is:

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

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

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