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: Set Up the Divisors
- Dividend: [tex]\(x^4 + 5x^3 - 3x - 15\)[/tex]
- 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].
- This gives us [tex]\(x\)[/tex].
### Step 3: Multiply and Subtract
- Multiply the entire divisor [tex]\(x^3 - 3\)[/tex] by [tex]\(x\)[/tex] to get [tex]\(x^4 - 3x\)[/tex].
- Subtract this result from the original dividend:
[tex]\[
\begin{align*}
(x^4 + 5x^3 - 3x - 15) - (x^4 - 3x) &= (x^4 + 5x^3 - 3x - 15) - x^4 + 3x \\
&= 5x^3 + 0x^2 - 3x + 3x - 15 \\
&= 5x^3 - 15
\end{align*}
\][/tex]
### Step 4: Repeat the Process
Now we divide the new leading term [tex]\(5x^3\)[/tex] by the leading term of the divisor [tex]\(x^3\)[/tex]:
- This gives us [tex]\(5\)[/tex].
- Multiply the entire divisor [tex]\(x^3 - 3\)[/tex] by [tex]\(5\)[/tex] to get [tex]\(5x^3 - 15\)[/tex].
### Step 5: Subtract Again
- Subtract [tex]\(5x^3 - 15\)[/tex] from [tex]\(5x^3 - 15\)[/tex]:
[tex]\[
(5x^3 - 15) - (5x^3 - 15) = 0
\][/tex]
Since there is no remainder left and we've used up all terms, the division process is complete. The quotient is the sum of the results from our steps which are [tex]\(x + 5\)[/tex].
Thus, 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 is:
[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: Set Up the Divisors
- Dividend: [tex]\(x^4 + 5x^3 - 3x - 15\)[/tex]
- 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].
- This gives us [tex]\(x\)[/tex].
### Step 3: Multiply and Subtract
- Multiply the entire divisor [tex]\(x^3 - 3\)[/tex] by [tex]\(x\)[/tex] to get [tex]\(x^4 - 3x\)[/tex].
- Subtract this result from the original dividend:
[tex]\[
\begin{align*}
(x^4 + 5x^3 - 3x - 15) - (x^4 - 3x) &= (x^4 + 5x^3 - 3x - 15) - x^4 + 3x \\
&= 5x^3 + 0x^2 - 3x + 3x - 15 \\
&= 5x^3 - 15
\end{align*}
\][/tex]
### Step 4: Repeat the Process
Now we divide the new leading term [tex]\(5x^3\)[/tex] by the leading term of the divisor [tex]\(x^3\)[/tex]:
- This gives us [tex]\(5\)[/tex].
- Multiply the entire divisor [tex]\(x^3 - 3\)[/tex] by [tex]\(5\)[/tex] to get [tex]\(5x^3 - 15\)[/tex].
### Step 5: Subtract Again
- Subtract [tex]\(5x^3 - 15\)[/tex] from [tex]\(5x^3 - 15\)[/tex]:
[tex]\[
(5x^3 - 15) - (5x^3 - 15) = 0
\][/tex]
Since there is no remainder left and we've used up all terms, the division process is complete. The quotient is the sum of the results from our steps which are [tex]\(x + 5\)[/tex].
Thus, 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 is:
[tex]\(x + 5\)[/tex]
