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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. Let's go through the steps:
### Step 1: Set Up the Division
Write the dividend [tex]\(x^4 + 5x^3 - 3x - 15\)[/tex] and the divisor [tex]\(x^3 - 3\)[/tex].
### 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], to get [tex]\(x\)[/tex].
### Step 3: Multiply and Subtract
- Multiply [tex]\(x\)[/tex] by the whole divisor [tex]\((x^3 - 3)\)[/tex]:
[tex]\[
x \cdot (x^3 - 3) = x^4 - 3x
\][/tex]
- Subtract this result from the original dividend:
[tex]\[
(x^4 + 5x^3 - 3x - 15) - (x^4 - 3x) = 5x^3 + 0x^2 - 0x - 15
\][/tex]
### Step 4: Repeat the Process
- The new dividend is [tex]\(5x^3 - 15\)[/tex].
- Divide the leading term [tex]\(5x^3\)[/tex] by [tex]\(x^3\)[/tex] to get [tex]\(5\)[/tex].
- Multiply and subtract:
[tex]\[
5 \cdot (x^3 - 3) = 5x^3 - 15
\][/tex]
[tex]\[
(5x^3 - 15) - (5x^3 - 15) = 0
\][/tex]
### Step 5: Write the Quotient
After the division, the quotient is the sum of the terms obtained from each step: [tex]\(x + 5\)[/tex].
Therefore, the quotient of the polynomial division is [tex]\(\boxed{x + 5}\)[/tex].
### Step 1: Set Up the Division
Write the dividend [tex]\(x^4 + 5x^3 - 3x - 15\)[/tex] and the divisor [tex]\(x^3 - 3\)[/tex].
### 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], to get [tex]\(x\)[/tex].
### Step 3: Multiply and Subtract
- Multiply [tex]\(x\)[/tex] by the whole divisor [tex]\((x^3 - 3)\)[/tex]:
[tex]\[
x \cdot (x^3 - 3) = x^4 - 3x
\][/tex]
- Subtract this result from the original dividend:
[tex]\[
(x^4 + 5x^3 - 3x - 15) - (x^4 - 3x) = 5x^3 + 0x^2 - 0x - 15
\][/tex]
### Step 4: Repeat the Process
- The new dividend is [tex]\(5x^3 - 15\)[/tex].
- Divide the leading term [tex]\(5x^3\)[/tex] by [tex]\(x^3\)[/tex] to get [tex]\(5\)[/tex].
- Multiply and subtract:
[tex]\[
5 \cdot (x^3 - 3) = 5x^3 - 15
\][/tex]
[tex]\[
(5x^3 - 15) - (5x^3 - 15) = 0
\][/tex]
### Step 5: Write the Quotient
After the division, the quotient is the sum of the terms obtained from each step: [tex]\(x + 5\)[/tex].
Therefore, the quotient of the polynomial division is [tex]\(\boxed{x + 5}\)[/tex].
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