Answer :
To find the quotient of the polynomial [tex]\((x^4 + 5x^3 - 3x - 15)\)[/tex] divided by [tex]\((x^3 - 3)\)[/tex], we can perform polynomial long division. Let's go through this process step-by-step:
1. Set Up the Problem:
- Dividend: [tex]\(x^4 + 5x^3 - 3x - 15\)[/tex]
- Divisor: [tex]\(x^3 - 3\)[/tex]
2. Perform Long Division:
- Look at the leading term of the dividend ([tex]\(x^4\)[/tex]) and the leading term of the divisor ([tex]\(x^3\)[/tex]).
- Divide the leading term of the dividend by the leading term of the divisor: [tex]\(\frac{x^4}{x^3} = x\)[/tex].
- This result, [tex]\(x\)[/tex], is part of the quotient.
3. Multiply and Subtract:
- Multiply the divisor [tex]\((x^3 - 3)\)[/tex] by [tex]\(x\)[/tex] to get [tex]\(x(x^3 - 3) = x^4 - 3x\)[/tex].
- Subtract this result from the original dividend:
[tex]\[
(x^4 + 5x^3 - 3x - 15) - (x^4 - 3x) = 5x^3 - 0x^2 - 3x - 15 + 3x
\][/tex]
- Simplify the expression to get the new dividend: [tex]\(5x^3 - 15\)[/tex].
4. Repeat the Process:
- Divide the leading term of the new dividend ([tex]\(5x^3\)[/tex]) by the leading term of the divisor ([tex]\(x^3\)[/tex]): [tex]\(\frac{5x^3}{x^3} = 5\)[/tex].
- This result, [tex]\(5\)[/tex], is the next term in the quotient.
- Multiply the divisor by [tex]\(5\)[/tex] to get [tex]\(5(x^3 - 3) = 5x^3 - 15\)[/tex].
- Subtract this from the new dividend:
[tex]\[
(5x^3 - 15) - (5x^3 - 15) = 0
\][/tex]
5. Conclusion:
- The remainder is [tex]\(0\)[/tex], and the division process is complete.
- 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 quotient is [tex]\(x + 5\)[/tex].
1. Set Up the Problem:
- Dividend: [tex]\(x^4 + 5x^3 - 3x - 15\)[/tex]
- Divisor: [tex]\(x^3 - 3\)[/tex]
2. Perform Long Division:
- Look at the leading term of the dividend ([tex]\(x^4\)[/tex]) and the leading term of the divisor ([tex]\(x^3\)[/tex]).
- Divide the leading term of the dividend by the leading term of the divisor: [tex]\(\frac{x^4}{x^3} = x\)[/tex].
- This result, [tex]\(x\)[/tex], is part of the quotient.
3. Multiply and Subtract:
- Multiply the divisor [tex]\((x^3 - 3)\)[/tex] by [tex]\(x\)[/tex] to get [tex]\(x(x^3 - 3) = x^4 - 3x\)[/tex].
- Subtract this result from the original dividend:
[tex]\[
(x^4 + 5x^3 - 3x - 15) - (x^4 - 3x) = 5x^3 - 0x^2 - 3x - 15 + 3x
\][/tex]
- Simplify the expression to get the new dividend: [tex]\(5x^3 - 15\)[/tex].
4. Repeat the Process:
- Divide the leading term of the new dividend ([tex]\(5x^3\)[/tex]) by the leading term of the divisor ([tex]\(x^3\)[/tex]): [tex]\(\frac{5x^3}{x^3} = 5\)[/tex].
- This result, [tex]\(5\)[/tex], is the next term in the quotient.
- Multiply the divisor by [tex]\(5\)[/tex] to get [tex]\(5(x^3 - 3) = 5x^3 - 15\)[/tex].
- Subtract this from the new dividend:
[tex]\[
(5x^3 - 15) - (5x^3 - 15) = 0
\][/tex]
5. Conclusion:
- The remainder is [tex]\(0\)[/tex], and the division process is complete.
- 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 quotient is [tex]\(x + 5\)[/tex].
