Answer :
Sure, let's find the quotient of [tex]\((x^4 + 5x^3 - 3x - 15)\)[/tex] divided by [tex]\((x^3 - 3)\)[/tex] using polynomial long division.
### Step-by-Step Solution
1. Set up the division:
We have the dividend [tex]\(x^4 + 5x^3 - 3x - 15\)[/tex] and the divisor [tex]\(x^3 - 3\)[/tex].
2. Divide the first term of the dividend by the first term of the divisor:
[tex]\[
\frac{x^4}{x^3} = x
\][/tex]
3. Multiply the entire divisor [tex]\(x^3 - 3\)[/tex] by this quotient term [tex]\(x\)[/tex]:
[tex]\[
x \cdot (x^3 - 3) = x^4 - 3x
\][/tex]
4. Subtract this result from the original dividend:
[tex]\[
(x^4 + 5x^3 - 3x - 15) - (x^4 - 3x) = 5x^3 + 0x^2 - 3x + 3x - 15 = 5x^3 - 15
\][/tex]
5. Repeat the process with the new polynomial [tex]\(5x^3 - 15\)[/tex]:
- Divide the first term of the new dividend [tex]\(5x^3\)[/tex] by the first term of the divisor [tex]\(x^3\)[/tex]:
[tex]\[
\frac{5x^3}{x^3} = 5
\][/tex]
- Multiply the entire divisor [tex]\(x^3 - 3\)[/tex] by this quotient term [tex]\(5\)[/tex]:
[tex]\[
5 \cdot (x^3 - 3) = 5x^3 - 15
\][/tex]
- Subtract this result from the new polynomial:
[tex]\[
(5x^3 - 15) - (5x^3 - 15) = 0
\][/tex]
6. Combine the quotient terms obtained from both steps:
The quotient from step 1 was [tex]\(x\)[/tex] and from step 5 was [tex]\(5\)[/tex], so combining them we get:
[tex]\[
\text{Quotient} = x + 5
\][/tex]
Thus, the quotient of [tex]\((x^4 + 5x^3 - 3x - 15)\)[/tex] divided by [tex]\((x^3 - 3)\)[/tex] is:
[tex]\[
\boxed{x + 5}
\][/tex]
### Step-by-Step Solution
1. Set up the division:
We have the dividend [tex]\(x^4 + 5x^3 - 3x - 15\)[/tex] and the divisor [tex]\(x^3 - 3\)[/tex].
2. Divide the first term of the dividend by the first term of the divisor:
[tex]\[
\frac{x^4}{x^3} = x
\][/tex]
3. Multiply the entire divisor [tex]\(x^3 - 3\)[/tex] by this quotient term [tex]\(x\)[/tex]:
[tex]\[
x \cdot (x^3 - 3) = x^4 - 3x
\][/tex]
4. Subtract this result from the original dividend:
[tex]\[
(x^4 + 5x^3 - 3x - 15) - (x^4 - 3x) = 5x^3 + 0x^2 - 3x + 3x - 15 = 5x^3 - 15
\][/tex]
5. Repeat the process with the new polynomial [tex]\(5x^3 - 15\)[/tex]:
- Divide the first term of the new dividend [tex]\(5x^3\)[/tex] by the first term of the divisor [tex]\(x^3\)[/tex]:
[tex]\[
\frac{5x^3}{x^3} = 5
\][/tex]
- Multiply the entire divisor [tex]\(x^3 - 3\)[/tex] by this quotient term [tex]\(5\)[/tex]:
[tex]\[
5 \cdot (x^3 - 3) = 5x^3 - 15
\][/tex]
- Subtract this result from the new polynomial:
[tex]\[
(5x^3 - 15) - (5x^3 - 15) = 0
\][/tex]
6. Combine the quotient terms obtained from both steps:
The quotient from step 1 was [tex]\(x\)[/tex] and from step 5 was [tex]\(5\)[/tex], so combining them we get:
[tex]\[
\text{Quotient} = x + 5
\][/tex]
Thus, the quotient of [tex]\((x^4 + 5x^3 - 3x - 15)\)[/tex] divided by [tex]\((x^3 - 3)\)[/tex] is:
[tex]\[
\boxed{x + 5}
\][/tex]
