Answer :
To find the quotient of the polynomial division [tex]\((x^4 + 5x^3 - 3x - 15) \div (x^3 - 3)\)[/tex], we perform polynomial long division.
### Step-by-Step 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 this term and subtract from the dividend:
[tex]\[
\text{Multiply: } x(x^3 - 3) = x^4 - 3x
\][/tex]
[tex]\[
\text{Subtract: } (x^4 + 5x^3 - 3x - 15) - (x^4 - 3x) = 5x^3 + 0x - 15
\][/tex]
3. Repeat the process with the new polynomial (remainder):
[tex]\[
\frac{5x^3}{x^3} = 5
\][/tex]
So, the next term of the quotient is [tex]\(5\)[/tex].
4. Multiply the entire divisor by this new term and subtract:
[tex]\[
\text{Multiply: } 5(x^3 - 3) = 5x^3 - 15
\][/tex]
[tex]\[
\text{Subtract: } (5x^3 - 15) - (5x^3 - 15) = 0
\][/tex]
The subtraction results in zero, indicating we have no remainder.
The polynomial division is complete, and the quotient is the sum of the terms obtained during the division, which is:
[tex]\[
x + 5
\][/tex]
Therefore, the quotient of [tex]\(\frac{x^4 + 5x^3 - 3x - 15}{x^3 - 3}\)[/tex] is [tex]\(x + 5\)[/tex].
Answer: [tex]\(x + 5\)[/tex]
### Step-by-Step 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 this term and subtract from the dividend:
[tex]\[
\text{Multiply: } x(x^3 - 3) = x^4 - 3x
\][/tex]
[tex]\[
\text{Subtract: } (x^4 + 5x^3 - 3x - 15) - (x^4 - 3x) = 5x^3 + 0x - 15
\][/tex]
3. Repeat the process with the new polynomial (remainder):
[tex]\[
\frac{5x^3}{x^3} = 5
\][/tex]
So, the next term of the quotient is [tex]\(5\)[/tex].
4. Multiply the entire divisor by this new term and subtract:
[tex]\[
\text{Multiply: } 5(x^3 - 3) = 5x^3 - 15
\][/tex]
[tex]\[
\text{Subtract: } (5x^3 - 15) - (5x^3 - 15) = 0
\][/tex]
The subtraction results in zero, indicating we have no remainder.
The polynomial division is complete, and the quotient is the sum of the terms obtained during the division, which is:
[tex]\[
x + 5
\][/tex]
Therefore, the quotient of [tex]\(\frac{x^4 + 5x^3 - 3x - 15}{x^3 - 3}\)[/tex] is [tex]\(x + 5\)[/tex].
Answer: [tex]\(x + 5\)[/tex]
