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 long division.
### Step-by-Step Solution:
1. Start Dividing:
- Divide the first term of the dividend, [tex]\(x^4\)[/tex], by the first term of the divisor, [tex]\(x^3\)[/tex].
- This gives us [tex]\(x\)[/tex] as the first term of the quotient.
2. Multiply and Subtract:
- Multiply the entire divisor [tex]\((x^3 - 3)\)[/tex] by [tex]\(x\)[/tex], which is the first term of the quotient. This gives [tex]\(x^4 - 3x\)[/tex].
- Subtract [tex]\(x^4 - 3x\)[/tex] from [tex]\((x^4 + 5x^3 - 3x - 15)\)[/tex]. The result is:
[tex]\[
(x^4 + 5x^3 - 3x - 15) - (x^4 - 3x) = 5x^3 + 0x^2 - 3x - 15
\][/tex]
- The [tex]\(x^4\)[/tex] terms cancel out.
3. Repeat the Division Process:
- Divide the new polynomial, [tex]\(5x^3 + 0x^2 - 3x - 15\)[/tex], by the divisor [tex]\(x^3 - 3\)[/tex].
- Divide [tex]\(5x^3\)[/tex] by [tex]\(x^3\)[/tex], which gives [tex]\(+5\)[/tex] as the next term of the quotient.
4. Multiply and Subtract Again:
- Multiply the divisor [tex]\((x^3 - 3)\)[/tex] by [tex]\(5\)[/tex], giving [tex]\(5x^3 - 15\)[/tex].
- Subtract [tex]\(5x^3 - 15\)[/tex] from the new polynomial:
[tex]\[
(5x^3 + 0x^2 - 3x - 15) - (5x^3 - 15) = 0x^3 + 0x^2 - 3x
\][/tex]
- The [tex]\(5x^3\)[/tex] terms cancel out.
5. Finalize the Quotient:
- No remaining polynomial terms of degree higher than or equal to the divisor's degree are left for further division.
- The division process is complete.
The quotient is thus [tex]\(x + 5\)[/tex].
Therefore, after the long division, 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 to the problem is [tex]\(x + 5\)[/tex].
### Step-by-Step Solution:
1. Start Dividing:
- Divide the first term of the dividend, [tex]\(x^4\)[/tex], by the first term of the divisor, [tex]\(x^3\)[/tex].
- This gives us [tex]\(x\)[/tex] as the first term of the quotient.
2. Multiply and Subtract:
- Multiply the entire divisor [tex]\((x^3 - 3)\)[/tex] by [tex]\(x\)[/tex], which is the first term of the quotient. This gives [tex]\(x^4 - 3x\)[/tex].
- Subtract [tex]\(x^4 - 3x\)[/tex] from [tex]\((x^4 + 5x^3 - 3x - 15)\)[/tex]. The result is:
[tex]\[
(x^4 + 5x^3 - 3x - 15) - (x^4 - 3x) = 5x^3 + 0x^2 - 3x - 15
\][/tex]
- The [tex]\(x^4\)[/tex] terms cancel out.
3. Repeat the Division Process:
- Divide the new polynomial, [tex]\(5x^3 + 0x^2 - 3x - 15\)[/tex], by the divisor [tex]\(x^3 - 3\)[/tex].
- Divide [tex]\(5x^3\)[/tex] by [tex]\(x^3\)[/tex], which gives [tex]\(+5\)[/tex] as the next term of the quotient.
4. Multiply and Subtract Again:
- Multiply the divisor [tex]\((x^3 - 3)\)[/tex] by [tex]\(5\)[/tex], giving [tex]\(5x^3 - 15\)[/tex].
- Subtract [tex]\(5x^3 - 15\)[/tex] from the new polynomial:
[tex]\[
(5x^3 + 0x^2 - 3x - 15) - (5x^3 - 15) = 0x^3 + 0x^2 - 3x
\][/tex]
- The [tex]\(5x^3\)[/tex] terms cancel out.
5. Finalize the Quotient:
- No remaining polynomial terms of degree higher than or equal to the divisor's degree are left for further division.
- The division process is complete.
The quotient is thus [tex]\(x + 5\)[/tex].
Therefore, after the long division, 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 to the problem is [tex]\(x + 5\)[/tex].
