Answer :
To solve the problem, we need to find the quotient when dividing the polynomial [tex]\(x^4 + 5x^3 - 3x - 15\)[/tex] by [tex]\(x^3 - 3\)[/tex].
To perform the division of polynomials, we'll use 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 our quotient is [tex]\(x\)[/tex].
2. Multiply the entire divisor by this term:
[tex]\[
x \cdot (x^3 - 3) = x^4 - 3x
\][/tex]
3. Subtract this product from the original polynomial:
[tex]\[
(x^4 + 5x^3 - 3x - 15) - (x^4 - 3x) = 5x^3 + 0x^2 - 3x + 15
\][/tex]
4. Repeat the process with the new polynomial:
- Divide the first term of the new polynomial by the first term of the divisor:
[tex]\[
\frac{5x^3}{x^3} = 5
\][/tex]
The next term of our quotient is [tex]\(5\)[/tex].
- Multiply the entire divisor by this term:
[tex]\[
5 \cdot (x^3 - 3) = 5x^3 - 15
\][/tex]
- Subtract this product from the new polynomial:
[tex]\[
(5x^3 + 0x^2 - 3x - 15) - (5x^3 - 15) = 0x^2 - 3x
\][/tex]
5. Since the degree of the remainder [tex]\(0x^2 - 3x\)[/tex] is less than the degree of the divisor [tex]\(x^3 - 3\)[/tex], the division process ends here.
So, the quotient of [tex]\(x^4 + 5x^3 - 3x - 15\)[/tex] divided by [tex]\(x^3 - 3\)[/tex] is [tex]\(x + 5\)[/tex], which is one of the choices provided in the list.
To perform the division of polynomials, we'll use 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 our quotient is [tex]\(x\)[/tex].
2. Multiply the entire divisor by this term:
[tex]\[
x \cdot (x^3 - 3) = x^4 - 3x
\][/tex]
3. Subtract this product from the original polynomial:
[tex]\[
(x^4 + 5x^3 - 3x - 15) - (x^4 - 3x) = 5x^3 + 0x^2 - 3x + 15
\][/tex]
4. Repeat the process with the new polynomial:
- Divide the first term of the new polynomial by the first term of the divisor:
[tex]\[
\frac{5x^3}{x^3} = 5
\][/tex]
The next term of our quotient is [tex]\(5\)[/tex].
- Multiply the entire divisor by this term:
[tex]\[
5 \cdot (x^3 - 3) = 5x^3 - 15
\][/tex]
- Subtract this product from the new polynomial:
[tex]\[
(5x^3 + 0x^2 - 3x - 15) - (5x^3 - 15) = 0x^2 - 3x
\][/tex]
5. Since the degree of the remainder [tex]\(0x^2 - 3x\)[/tex] is less than the degree of the divisor [tex]\(x^3 - 3\)[/tex], the division process ends here.
So, the quotient of [tex]\(x^4 + 5x^3 - 3x - 15\)[/tex] divided by [tex]\(x^3 - 3\)[/tex] is [tex]\(x + 5\)[/tex], which is one of the choices provided in the list.
