Answer :
To find the quotient of the polynomial [tex]\((x^4 + 5x^3 - 3x - 15)\)[/tex] divided by [tex]\((x^3 - 3)\)[/tex], we need to perform polynomial division. Here’s a step-by-step breakdown of the process:
1. Set Up Division:
- We're dividing the polynomial [tex]\(x^4 + 5x^3 - 3x - 15\)[/tex] (the dividend) by [tex]\(x^3 - 3\)[/tex] (the divisor).
2. Determine the Leading Terms:
- The leading term in the dividend is [tex]\(x^4\)[/tex], and the leading term in the divisor is [tex]\(x^3\)[/tex].
3. Divide Leading Terms:
- 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]
4. Multiply and Subtract:
- Multiply the entire divisor by this result: [tex]\(x \times (x^3 - 3) = x^4 - 3x\)[/tex].
- Subtract this from the original dividend:
[tex]\[
(x^4 + 5x^3 - 3x - 15) - (x^4 - 3x) = 5x^3 + 3x - 15.
\][/tex]
5. Repeat the Process:
- Now, divide the new 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 5: [tex]\(5(x^3 - 3) = 5x^3 - 15\)[/tex].
- Subtract this from the preceding result:
[tex]\[
(5x^3 + 3x - 15) - (5x^3 - 15) = 3x.
\][/tex]
6. Check for More Steps:
- There are no more terms to divide since the degree of the remainder [tex]\(3x\)[/tex] is less than the degree of the divisor [tex]\(x^3 - 3\)[/tex].
7. Assemble the Quotient:
- The quotient is [tex]\(x + 5\)[/tex].
- The remainder is [tex]\(0\)[/tex], confirming that division results in an exact polynomial quotient.
Thus, the quotient of [tex]\((x^4 + 5x^3 - 3x - 15)\)[/tex] divided by [tex]\((x^3 - 3)\)[/tex] is [tex]\(x + 5\)[/tex].
This means that the correct quotient is [tex]\(x + 5\)[/tex], which matches one of the given options.
1. Set Up Division:
- We're dividing the polynomial [tex]\(x^4 + 5x^3 - 3x - 15\)[/tex] (the dividend) by [tex]\(x^3 - 3\)[/tex] (the divisor).
2. Determine the Leading Terms:
- The leading term in the dividend is [tex]\(x^4\)[/tex], and the leading term in the divisor is [tex]\(x^3\)[/tex].
3. Divide Leading Terms:
- 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]
4. Multiply and Subtract:
- Multiply the entire divisor by this result: [tex]\(x \times (x^3 - 3) = x^4 - 3x\)[/tex].
- Subtract this from the original dividend:
[tex]\[
(x^4 + 5x^3 - 3x - 15) - (x^4 - 3x) = 5x^3 + 3x - 15.
\][/tex]
5. Repeat the Process:
- Now, divide the new 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 5: [tex]\(5(x^3 - 3) = 5x^3 - 15\)[/tex].
- Subtract this from the preceding result:
[tex]\[
(5x^3 + 3x - 15) - (5x^3 - 15) = 3x.
\][/tex]
6. Check for More Steps:
- There are no more terms to divide since the degree of the remainder [tex]\(3x\)[/tex] is less than the degree of the divisor [tex]\(x^3 - 3\)[/tex].
7. Assemble the Quotient:
- The quotient is [tex]\(x + 5\)[/tex].
- The remainder is [tex]\(0\)[/tex], confirming that division results in an exact polynomial quotient.
Thus, the quotient of [tex]\((x^4 + 5x^3 - 3x - 15)\)[/tex] divided by [tex]\((x^3 - 3)\)[/tex] is [tex]\(x + 5\)[/tex].
This means that the correct quotient is [tex]\(x + 5\)[/tex], which matches one of the given options.
