Answer :
Sure! Let's solve this problem step-by-step.
We need to find the quotient of the polynomial division [tex]\((x^4 + 5x^3 - 3x - 15) \div (x^3 - 3)\)[/tex].
### Step-by-Step Solution:
1. Setup the Division: Write down the polynomials in a long division format with [tex]\(x^4 + 5x^3 - 3x - 15\)[/tex] as the dividend and [tex]\(x^3 - 3\)[/tex] as the divisor.
2. First Division: Divide the leading term of the dividend [tex]\(x^4\)[/tex] by the leading term of the divisor [tex]\(x^3\)[/tex]:
[tex]\[
\frac{x^4}{x^3} = x
\][/tex]
Multiply the entire divisor by [tex]\(x\)[/tex] and subtract from the dividend:
[tex]\[
(x^4 + 5x^3 - 3x - 15) - (x \cdot (x^3 - 3)) = (x^4 + 5x^3 - 3x - 15) - (x^4 - 3x) = 5x^3 + 0x - 15
\][/tex]
3. Second Division: Now, take the new polynomial [tex]\(5x^3 - 3x - 15\)[/tex]. Divide the leading term [tex]\(5x^3\)[/tex] by the leading term of the divisor [tex]\(x^3\)[/tex]:
[tex]\[
\frac{5x^3}{x^3} = 5
\][/tex]
Multiply the entire divisor by [tex]\(5\)[/tex] and subtract from the new polynomial:
[tex]\[
(5x^3 - 3x - 15) - (5 \cdot (x^3 - 3)) = (5x^3 - 3x - 15) - (5x^3 - 15) = -3x
\][/tex]
4. Conclusion: Since the degree of the resulting polynomial [tex]\(-3x\)[/tex] is less than the degree of [tex]\(x^3 - 3\)[/tex], the division stops here. The quotient is the combination of the terms we've found in each division step:
[tex]\[
x + 5
\][/tex]
### Final Answer:
The quotient of [tex]\((x^4 + 5x^3 - 3x - 15)\)[/tex] divided by [tex]\((x^3 - 3)\)[/tex] is:
[tex]\[
x + 5
\][/tex]
So, the correct answer is [tex]\(x + 5\)[/tex].
We need to find the quotient of the polynomial division [tex]\((x^4 + 5x^3 - 3x - 15) \div (x^3 - 3)\)[/tex].
### Step-by-Step Solution:
1. Setup the Division: Write down the polynomials in a long division format with [tex]\(x^4 + 5x^3 - 3x - 15\)[/tex] as the dividend and [tex]\(x^3 - 3\)[/tex] as the divisor.
2. First Division: Divide the leading term of the dividend [tex]\(x^4\)[/tex] by the leading term of the divisor [tex]\(x^3\)[/tex]:
[tex]\[
\frac{x^4}{x^3} = x
\][/tex]
Multiply the entire divisor by [tex]\(x\)[/tex] and subtract from the dividend:
[tex]\[
(x^4 + 5x^3 - 3x - 15) - (x \cdot (x^3 - 3)) = (x^4 + 5x^3 - 3x - 15) - (x^4 - 3x) = 5x^3 + 0x - 15
\][/tex]
3. Second Division: Now, take the new polynomial [tex]\(5x^3 - 3x - 15\)[/tex]. Divide the leading term [tex]\(5x^3\)[/tex] by the leading term of the divisor [tex]\(x^3\)[/tex]:
[tex]\[
\frac{5x^3}{x^3} = 5
\][/tex]
Multiply the entire divisor by [tex]\(5\)[/tex] and subtract from the new polynomial:
[tex]\[
(5x^3 - 3x - 15) - (5 \cdot (x^3 - 3)) = (5x^3 - 3x - 15) - (5x^3 - 15) = -3x
\][/tex]
4. Conclusion: Since the degree of the resulting polynomial [tex]\(-3x\)[/tex] is less than the degree of [tex]\(x^3 - 3\)[/tex], the division stops here. The quotient is the combination of the terms we've found in each division step:
[tex]\[
x + 5
\][/tex]
### Final Answer:
The quotient of [tex]\((x^4 + 5x^3 - 3x - 15)\)[/tex] divided by [tex]\((x^3 - 3)\)[/tex] is:
[tex]\[
x + 5
\][/tex]
So, the correct answer is [tex]\(x + 5\)[/tex].
