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].
### 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].
