Answer :
To find the quotient of the polynomial division of [tex]\(x^4 + 5x^3 - 3x - 15\)[/tex] by [tex]\(x^3 - 3\)[/tex], we will 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]
2. Multiply the whole divisor by this result:
[tex]\[
x \cdot (x^3 - 3) = x^4 - 3x
\][/tex]
3. Subtract the result from the original dividend:
[tex]\[
(x^4 + 5x^3 - 3x - 15) - (x^4 - 3x) = 5x^3 + 0 - 15
\][/tex]
4. Bring down the next term (if required):
The subtraction resulted in [tex]\(5x^3 - 15\)[/tex].
5. Repeat the process:
- Divide the first term [tex]\(5x^3\)[/tex] by [tex]\(x^3\)[/tex]:
[tex]\[
\frac{5x^3}{x^3} = 5
\][/tex]
- Multiply the entire divisor by this result:
[tex]\[
5 \cdot (x^3 - 3) = 5x^3 - 15
\][/tex]
- Subtract this from what is left:
[tex]\[
(5x^3 - 15) - (5x^3 - 15) = 0
\][/tex]
The remainder is 0 after subtraction, indicating that the division is exact, and the quotient is simply the result from step 1 and step 5:
The quotient is:
[tex]\[
x + 5
\][/tex]
So, the correct answer is [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]
2. Multiply the whole divisor by this result:
[tex]\[
x \cdot (x^3 - 3) = x^4 - 3x
\][/tex]
3. Subtract the result from the original dividend:
[tex]\[
(x^4 + 5x^3 - 3x - 15) - (x^4 - 3x) = 5x^3 + 0 - 15
\][/tex]
4. Bring down the next term (if required):
The subtraction resulted in [tex]\(5x^3 - 15\)[/tex].
5. Repeat the process:
- Divide the first term [tex]\(5x^3\)[/tex] by [tex]\(x^3\)[/tex]:
[tex]\[
\frac{5x^3}{x^3} = 5
\][/tex]
- Multiply the entire divisor by this result:
[tex]\[
5 \cdot (x^3 - 3) = 5x^3 - 15
\][/tex]
- Subtract this from what is left:
[tex]\[
(5x^3 - 15) - (5x^3 - 15) = 0
\][/tex]
The remainder is 0 after subtraction, indicating that the division is exact, and the quotient is simply the result from step 1 and step 5:
The quotient is:
[tex]\[
x + 5
\][/tex]
So, the correct answer is [tex]\(x + 5\)[/tex].
