Answer :
To find the quotient of [tex]\((x^4 + 5x^3 - 3x - 15)\)[/tex] divided by [tex]\((x^3 - 3)\)[/tex], we can perform polynomial long division. Here's a step-by-step solution:
### Step 1: Division Setup
Set up the division:
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]):
[tex]\[ \frac{x^4}{x^3} = x \][/tex]
This [tex]\(x\)[/tex] is the first term of the quotient.
### Step 3: Multiply and Subtract
Multiply the entire divisor by [tex]\(x\)[/tex] and subtract the result from the dividend:
[tex]\[
(x^3 - 3) \times x = x^4 - 3x
\][/tex]
Subtract from the original polynomial:
[tex]\[
(x^4 + 5x^3 - 3x - 15) - (x^4 - 3x) = 5x^3 + 0x^2 - 15
\][/tex]
### Step 4: Repeat the Process
Divide the new leading term ([tex]\(5x^3\)[/tex]) by [tex]\(x^3\)[/tex]:
[tex]\[ \frac{5x^3}{x^3} = 5 \][/tex]
This 5 is the next term in the quotient.
Multiply and subtract:
[tex]\[
5 \times (x^3 - 3) = 5x^3 - 15
\][/tex]
Subtract:
[tex]\[
(5x^3 + 0x^2 - 15) - (5x^3 - 15) = 0
\][/tex]
### Conclusion
After the division process, the remainder is 0, which means the quotient is simply:
[tex]\(x + 5\)[/tex]
So the quotient of [tex]\((x^4+5x^3-3x-15)\)[/tex] divided by [tex]\((x^3-3)\)[/tex] is [tex]\(x + 5\)[/tex].
### Step 1: Division Setup
Set up the division:
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]):
[tex]\[ \frac{x^4}{x^3} = x \][/tex]
This [tex]\(x\)[/tex] is the first term of the quotient.
### Step 3: Multiply and Subtract
Multiply the entire divisor by [tex]\(x\)[/tex] and subtract the result from the dividend:
[tex]\[
(x^3 - 3) \times x = x^4 - 3x
\][/tex]
Subtract from the original polynomial:
[tex]\[
(x^4 + 5x^3 - 3x - 15) - (x^4 - 3x) = 5x^3 + 0x^2 - 15
\][/tex]
### Step 4: Repeat the Process
Divide the new leading term ([tex]\(5x^3\)[/tex]) by [tex]\(x^3\)[/tex]:
[tex]\[ \frac{5x^3}{x^3} = 5 \][/tex]
This 5 is the next term in the quotient.
Multiply and subtract:
[tex]\[
5 \times (x^3 - 3) = 5x^3 - 15
\][/tex]
Subtract:
[tex]\[
(5x^3 + 0x^2 - 15) - (5x^3 - 15) = 0
\][/tex]
### Conclusion
After the division process, the remainder is 0, which means the quotient is simply:
[tex]\(x + 5\)[/tex]
So the quotient of [tex]\((x^4+5x^3-3x-15)\)[/tex] divided by [tex]\((x^3-3)\)[/tex] is [tex]\(x + 5\)[/tex].
