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Answer :
To solve the problem of finding the quotient of the polynomial division [tex]\( \frac{x^4 + 5x^3 - 3x - 15}{x^3 - 3} \)[/tex], we can perform polynomial long division. Here's a step-by-step guide:
### Step 1: Set up the division
We'll divide the polynomial [tex]\( x^4 + 5x^3 - 3x - 15 \)[/tex] (the dividend) by [tex]\( x^3 - 3 \)[/tex] (the divisor).
### Step 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].
- [tex]\( x^4 \div x^3 = x \)[/tex].
This quotient term, [tex]\( x \)[/tex], will be the first term of our result.
### Step 3: Multiply and subtract
- Multiply the entire divisor [tex]\( x^3 - 3 \)[/tex] by [tex]\( x \)[/tex] (the quotient term we just found):
- [tex]\( x(x^3 - 3) = x^4 - 3x \)[/tex].
- Subtract this result from the original dividend:
- [tex]\( (x^4 + 5x^3 - 3x - 15) - (x^4 - 3x) = 5x^3 + 0x^2 - 3x + 0x - 15 + 3x \)[/tex].
- Simplifying gives: [tex]\( 5x^3 + 3x - 15 \)[/tex].
### Step 4: Repeat the process
- Divide the new leading term [tex]\( 5x^3 \)[/tex] by the leading term of the divisor, [tex]\( x^3 \)[/tex]:
- [tex]\( 5x^3 \div x^3 = 5 \)[/tex].
- Multiply the entire divisor by this new quotient term:
- [tex]\( 5(x^3 - 3) = 5x^3 - 15 \)[/tex].
- Subtract this result from the current polynomial:
- [tex]\( (5x^3 + 3x - 15) - (5x^3 - 15) = 3x \)[/tex].
### Final Result
Now, the remainder is [tex]\( 3x \)[/tex], and since it is of lower degree than the divisor, we stop here.
Thus, the quotient is [tex]\( x + 5 \)[/tex].
The quotient of the polynomial division [tex]\( \frac{x^4 + 5x^3 - 3x - 15}{x^3 - 3} \)[/tex] is [tex]\(\boxed{x + 5}\)[/tex].
### Step 1: Set up the division
We'll divide the polynomial [tex]\( x^4 + 5x^3 - 3x - 15 \)[/tex] (the dividend) by [tex]\( x^3 - 3 \)[/tex] (the divisor).
### Step 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].
- [tex]\( x^4 \div x^3 = x \)[/tex].
This quotient term, [tex]\( x \)[/tex], will be the first term of our result.
### Step 3: Multiply and subtract
- Multiply the entire divisor [tex]\( x^3 - 3 \)[/tex] by [tex]\( x \)[/tex] (the quotient term we just found):
- [tex]\( x(x^3 - 3) = x^4 - 3x \)[/tex].
- Subtract this result from the original dividend:
- [tex]\( (x^4 + 5x^3 - 3x - 15) - (x^4 - 3x) = 5x^3 + 0x^2 - 3x + 0x - 15 + 3x \)[/tex].
- Simplifying gives: [tex]\( 5x^3 + 3x - 15 \)[/tex].
### Step 4: Repeat the process
- Divide the new leading term [tex]\( 5x^3 \)[/tex] by the leading term of the divisor, [tex]\( x^3 \)[/tex]:
- [tex]\( 5x^3 \div x^3 = 5 \)[/tex].
- Multiply the entire divisor by this new quotient term:
- [tex]\( 5(x^3 - 3) = 5x^3 - 15 \)[/tex].
- Subtract this result from the current polynomial:
- [tex]\( (5x^3 + 3x - 15) - (5x^3 - 15) = 3x \)[/tex].
### Final Result
Now, the remainder is [tex]\( 3x \)[/tex], and since it is of lower degree than the divisor, we stop here.
Thus, the quotient is [tex]\( x + 5 \)[/tex].
The quotient of the polynomial division [tex]\( \frac{x^4 + 5x^3 - 3x - 15}{x^3 - 3} \)[/tex] is [tex]\(\boxed{x + 5}\)[/tex].
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