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Answer :
Let's solve the problem step-by-step by performing polynomial long division.
We need to divide the polynomial [tex]\(x^4 + 5x^3 - 3x - 15\)[/tex] by [tex]\(x^3 - 3\)[/tex].
### Step 1: Setup
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]:
[tex]\[
\frac{x^4}{x^3} = x
\][/tex]
### Step 3: Multiply and Subtract
Multiply the entire divisor [tex]\(x^3 - 3\)[/tex] by the result from Step 2 ([tex]\(x\)[/tex]):
[tex]\[
x(x^3 - 3) = x^4 - 3x
\][/tex]
Subtract this result from the original dividend:
[tex]\[
(x^4 + 5x^3 - 3x - 15) - (x^4 - 3x) = 5x^3 + 0x^2 - 15
\][/tex]
### Step 4: Repeat
Now repeat the process with the new polynomial [tex]\(5x^3 + 0x^2 - 15\)[/tex].
- Divide the leading term [tex]\(5x^3\)[/tex] by [tex]\(x^3\)[/tex]:
[tex]\[
\frac{5x^3}{x^3} = 5
\][/tex]
- Multiply the divisor by [tex]\(5\)[/tex]:
[tex]\[
5(x^3 - 3) = 5x^3 - 15
\][/tex]
- Subtract the result from [tex]\(5x^3 + 0x^2 - 15\)[/tex]:
[tex]\[
(5x^3 + 0x^2 - 15) - (5x^3 - 15) = 0x^2 + 0x
\][/tex]
Since we are left with a zero remainder, we stop here.
### Final Quotient
The quotient of the division is the terms we found during the division steps:
[tex]\[
x + 5
\][/tex]
So, the quotient of [tex]\(\left(x^4 + 5x^3 - 3x - 15\right)\)[/tex] and [tex]\(\left(x^3 - 3\right)\)[/tex] is [tex]\(x + 5\)[/tex].
Answer: [tex]\(x + 5\)[/tex]
We need to divide the polynomial [tex]\(x^4 + 5x^3 - 3x - 15\)[/tex] by [tex]\(x^3 - 3\)[/tex].
### Step 1: Setup
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]:
[tex]\[
\frac{x^4}{x^3} = x
\][/tex]
### Step 3: Multiply and Subtract
Multiply the entire divisor [tex]\(x^3 - 3\)[/tex] by the result from Step 2 ([tex]\(x\)[/tex]):
[tex]\[
x(x^3 - 3) = x^4 - 3x
\][/tex]
Subtract this result from the original dividend:
[tex]\[
(x^4 + 5x^3 - 3x - 15) - (x^4 - 3x) = 5x^3 + 0x^2 - 15
\][/tex]
### Step 4: Repeat
Now repeat the process with the new polynomial [tex]\(5x^3 + 0x^2 - 15\)[/tex].
- Divide the leading term [tex]\(5x^3\)[/tex] by [tex]\(x^3\)[/tex]:
[tex]\[
\frac{5x^3}{x^3} = 5
\][/tex]
- Multiply the divisor by [tex]\(5\)[/tex]:
[tex]\[
5(x^3 - 3) = 5x^3 - 15
\][/tex]
- Subtract the result from [tex]\(5x^3 + 0x^2 - 15\)[/tex]:
[tex]\[
(5x^3 + 0x^2 - 15) - (5x^3 - 15) = 0x^2 + 0x
\][/tex]
Since we are left with a zero remainder, we stop here.
### Final Quotient
The quotient of the division is the terms we found during the division steps:
[tex]\[
x + 5
\][/tex]
So, the quotient of [tex]\(\left(x^4 + 5x^3 - 3x - 15\right)\)[/tex] and [tex]\(\left(x^3 - 3\right)\)[/tex] is [tex]\(x + 5\)[/tex].
Answer: [tex]\(x + 5\)[/tex]
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