Answer :
To find the quotient of the polynomial division [tex]\((x^4 + 5x^3 - 3x - 15)\)[/tex] by [tex]\((x^3 - 3)\)[/tex], we perform polynomial long division.
Here's a step-by-step guide on how to do it:
1. Setup the Division:
- Dividend: [tex]\(x^4 + 5x^3 - 3x - 15\)[/tex]
- Divisor: [tex]\(x^3 - 3\)[/tex]
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 you the first term of the quotient: [tex]\(x\)[/tex].
3. Multiply and Subtract:
- Multiply the entire divisor ([tex]\(x^3 - 3\)[/tex]) by the term [tex]\(x\)[/tex], resulting in [tex]\(x^4 - 3x\)[/tex].
- Subtract this result from the original dividend:
[tex]\[
(x^4 + 5x^3 - 3x - 15) - (x^4 - 3x) = 5x^3 + 3x - 15
\][/tex]
4. Repeat the Process:
- Now, divide the leading term of the new dividend ([tex]\(5x^3\)[/tex]) by the leading term of the divisor ([tex]\(x^3\)[/tex]).
- This gives you the next term of the quotient: [tex]\(+5\)[/tex].
5. Multiply and Subtract Again:
- Multiply the entire divisor ([tex]\(x^3 - 3\)[/tex]) by [tex]\(5\)[/tex], resulting in [tex]\(5x^3 - 15\)[/tex].
- Subtract this from the current dividend:
[tex]\[
(5x^3 + 3x - 15) - (5x^3 - 15) = 3x
\][/tex]
6. End Point:
- Since the remainder [tex]\(3x\)[/tex] has a degree ([tex]\(1\)[/tex]) less than the degree of the divisor ([tex]\(3\)[/tex]), the process stops here.
7. Final Quotient:
- The quotient is the result obtained from adding the terms you have calculated: [tex]\(x + 5\)[/tex].
So, the quotient of dividing [tex]\((x^4 + 5x^3 - 3x - 15)\)[/tex] by [tex]\((x^3 - 3)\)[/tex] is [tex]\((x + 5)\)[/tex].
Here's a step-by-step guide on how to do it:
1. Setup the Division:
- Dividend: [tex]\(x^4 + 5x^3 - 3x - 15\)[/tex]
- Divisor: [tex]\(x^3 - 3\)[/tex]
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 you the first term of the quotient: [tex]\(x\)[/tex].
3. Multiply and Subtract:
- Multiply the entire divisor ([tex]\(x^3 - 3\)[/tex]) by the term [tex]\(x\)[/tex], resulting in [tex]\(x^4 - 3x\)[/tex].
- Subtract this result from the original dividend:
[tex]\[
(x^4 + 5x^3 - 3x - 15) - (x^4 - 3x) = 5x^3 + 3x - 15
\][/tex]
4. Repeat the Process:
- Now, divide the leading term of the new dividend ([tex]\(5x^3\)[/tex]) by the leading term of the divisor ([tex]\(x^3\)[/tex]).
- This gives you the next term of the quotient: [tex]\(+5\)[/tex].
5. Multiply and Subtract Again:
- Multiply the entire divisor ([tex]\(x^3 - 3\)[/tex]) by [tex]\(5\)[/tex], resulting in [tex]\(5x^3 - 15\)[/tex].
- Subtract this from the current dividend:
[tex]\[
(5x^3 + 3x - 15) - (5x^3 - 15) = 3x
\][/tex]
6. End Point:
- Since the remainder [tex]\(3x\)[/tex] has a degree ([tex]\(1\)[/tex]) less than the degree of the divisor ([tex]\(3\)[/tex]), the process stops here.
7. Final Quotient:
- The quotient is the result obtained from adding the terms you have calculated: [tex]\(x + 5\)[/tex].
So, the quotient of dividing [tex]\((x^4 + 5x^3 - 3x - 15)\)[/tex] by [tex]\((x^3 - 3)\)[/tex] is [tex]\((x + 5)\)[/tex].
