Answer :
To find the quotient of [tex]\((x^4 + 5x^3 - 3x - 15)\)[/tex] divided by [tex]\((x^3 - 3)\)[/tex], we can perform polynomial division. Here's a step-by-step guide:
1. Set up the division:
Divide the polynomial [tex]\(x^4 + 5x^3 - 3x - 15\)[/tex] by the polynomial [tex]\(x^3 - 3\)[/tex].
2. Divide the leading terms:
- Divide the leading term of the dividend [tex]\(x^4\)[/tex] by the leading term of the divisor [tex]\(x^3\)[/tex], which gives [tex]\(x\)[/tex].
3. Multiply and subtract:
- Multiply the entire divisor [tex]\(x^3 - 3\)[/tex] by [tex]\(x\)[/tex] to get [tex]\(x \cdot (x^3 - 3) = x^4 - 3x\)[/tex].
- Subtract [tex]\(x^4 - 3x\)[/tex] from [tex]\(x^4 + 5x^3 - 3x - 15\)[/tex] to get:
[tex]\[
(x^4 + 5x^3 - 3x - 15) - (x^4 - 3x) = 5x^3 + 3x - 15
\][/tex]
4. Repeat the process:
- Divide the new leading term [tex]\(5x^3\)[/tex] by [tex]\(x^3\)[/tex], which gives [tex]\(5\)[/tex].
- Multiply the divisor [tex]\(x^3 - 3\)[/tex] by [tex]\(5\)[/tex] to get [tex]\(5x^3 - 15\)[/tex].
- Subtract [tex]\(5x^3 - 15\)[/tex] from [tex]\(5x^3 + 3x - 15\)[/tex] to get:
[tex]\[
(5x^3 + 3x - 15) - (5x^3 - 15) = 3x
\][/tex]
5. Terminate the process:
- The degree of the resulting polynomial [tex]\(3x\)[/tex] is lower than the degree of the divisor [tex]\(x^3 - 3\)[/tex], which means the division process stops here.
- The result is the quotient [tex]\(x + 5\)[/tex] with a remainder of [tex]\(3x\)[/tex].
Since we're looking for the quotient, the answer is [tex]\(x + 5\)[/tex].
From the choices given, the correct answer is:
- [tex]\(x + 5\)[/tex]
1. Set up the division:
Divide the polynomial [tex]\(x^4 + 5x^3 - 3x - 15\)[/tex] by the polynomial [tex]\(x^3 - 3\)[/tex].
2. Divide the leading terms:
- Divide the leading term of the dividend [tex]\(x^4\)[/tex] by the leading term of the divisor [tex]\(x^3\)[/tex], which gives [tex]\(x\)[/tex].
3. Multiply and subtract:
- Multiply the entire divisor [tex]\(x^3 - 3\)[/tex] by [tex]\(x\)[/tex] to get [tex]\(x \cdot (x^3 - 3) = x^4 - 3x\)[/tex].
- Subtract [tex]\(x^4 - 3x\)[/tex] from [tex]\(x^4 + 5x^3 - 3x - 15\)[/tex] to get:
[tex]\[
(x^4 + 5x^3 - 3x - 15) - (x^4 - 3x) = 5x^3 + 3x - 15
\][/tex]
4. Repeat the process:
- Divide the new leading term [tex]\(5x^3\)[/tex] by [tex]\(x^3\)[/tex], which gives [tex]\(5\)[/tex].
- Multiply the divisor [tex]\(x^3 - 3\)[/tex] by [tex]\(5\)[/tex] to get [tex]\(5x^3 - 15\)[/tex].
- Subtract [tex]\(5x^3 - 15\)[/tex] from [tex]\(5x^3 + 3x - 15\)[/tex] to get:
[tex]\[
(5x^3 + 3x - 15) - (5x^3 - 15) = 3x
\][/tex]
5. Terminate the process:
- The degree of the resulting polynomial [tex]\(3x\)[/tex] is lower than the degree of the divisor [tex]\(x^3 - 3\)[/tex], which means the division process stops here.
- The result is the quotient [tex]\(x + 5\)[/tex] with a remainder of [tex]\(3x\)[/tex].
Since we're looking for the quotient, the answer is [tex]\(x + 5\)[/tex].
From the choices given, the correct answer is:
- [tex]\(x + 5\)[/tex]
