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Answer :
Let's solve the problem by performing polynomial long division to find the quotient of [tex]\( \frac{x^4 + 5x^3 - 3x - 15}{x^3 - 3} \)[/tex].
### Step-by-step Solution:
1. Setup the Division:
- Dividend: [tex]\( x^4 + 5x^3 - 3x - 15 \)[/tex]
- Divisor: [tex]\( x^3 - 3 \)[/tex]
2. Divide the Leading Terms:
- Divide [tex]\( x^4 \)[/tex] by [tex]\( x^3 \)[/tex], which gives us [tex]\( x \)[/tex].
3. Multiply and Subtract:
- Multiply [tex]\( x \)[/tex] by the divisor: [tex]\( x \cdot (x^3 - 3) = x^4 - 3x \)[/tex].
- Subtract this result from the dividend:
[tex]\[
(x^4 + 5x^3 - 3x - 15) - (x^4 - 3x) = 5x^3 + 0x^2 + 0 - 15
\][/tex]
4. Bring Down the Next Term:
- The result from subtraction is [tex]\( 5x^3 - 15 \)[/tex].
5. Repeat the Division Process:
- Divide the new leading term [tex]\( 5x^3 \)[/tex] by [tex]\( x^3 \)[/tex], which gives us [tex]\( 5 \)[/tex].
6. Multiply and Subtract Again:
- Multiply [tex]\( 5 \)[/tex] by the divisor: [tex]\( 5 \cdot (x^3 - 3) = 5x^3 - 15 \)[/tex].
- Subtract this result from the current expression:
[tex]\[
(5x^3 - 15) - (5x^3 - 15) = 0
\][/tex]
7. No Remainder:
- The remainder is 0, which means the division is exact.
### Conclusion:
The quotient is [tex]\( x + 5 \)[/tex].
Therefore, the correct answer is:
C) [tex]\( x + 5 \)[/tex]
### Step-by-step Solution:
1. Setup the Division:
- Dividend: [tex]\( x^4 + 5x^3 - 3x - 15 \)[/tex]
- Divisor: [tex]\( x^3 - 3 \)[/tex]
2. Divide the Leading Terms:
- Divide [tex]\( x^4 \)[/tex] by [tex]\( x^3 \)[/tex], which gives us [tex]\( x \)[/tex].
3. Multiply and Subtract:
- Multiply [tex]\( x \)[/tex] by the divisor: [tex]\( x \cdot (x^3 - 3) = x^4 - 3x \)[/tex].
- Subtract this result from the dividend:
[tex]\[
(x^4 + 5x^3 - 3x - 15) - (x^4 - 3x) = 5x^3 + 0x^2 + 0 - 15
\][/tex]
4. Bring Down the Next Term:
- The result from subtraction is [tex]\( 5x^3 - 15 \)[/tex].
5. Repeat the Division Process:
- Divide the new leading term [tex]\( 5x^3 \)[/tex] by [tex]\( x^3 \)[/tex], which gives us [tex]\( 5 \)[/tex].
6. Multiply and Subtract Again:
- Multiply [tex]\( 5 \)[/tex] by the divisor: [tex]\( 5 \cdot (x^3 - 3) = 5x^3 - 15 \)[/tex].
- Subtract this result from the current expression:
[tex]\[
(5x^3 - 15) - (5x^3 - 15) = 0
\][/tex]
7. No Remainder:
- The remainder is 0, which means the division is exact.
### Conclusion:
The quotient is [tex]\( x + 5 \)[/tex].
Therefore, the correct answer is:
C) [tex]\( x + 5 \)[/tex]
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