Answer :
To solve the problem, we need to divide the polynomial [tex]\((x^4 + 5x^3 - 3x - 15)\)[/tex] by the polynomial [tex]\((x^3 - 3)\)[/tex] and find the quotient.
Here's how you can perform the division step-by-step:
1. Set Up the Division:
Write the dividend (the polynomial being divided) [tex]\((x^4 + 5x^3 - 3x - 15)\)[/tex] and the 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].
- [tex]\(x^4 \div x^3 = x\)[/tex]. This is the first term of the quotient.
3. Multiply and Subtract:
- Multiply [tex]\((x^3 - 3)\)[/tex] by the first term of the quotient we found, which is [tex]\(x\)[/tex].
- This gives: [tex]\(x(x^3 - 3) = x^4 - 3x\)[/tex].
- Subtract this from the original dividend: [tex]\((x^4 + 5x^3 - 3x - 15) - (x^4 - 3x) = 5x^3 - 15\)[/tex].
4. Repeat the Process:
- Divide the new leading term, [tex]\(5x^3\)[/tex], by the leading term of the divisor, [tex]\(x^3\)[/tex].
- [tex]\(5x^3 \div x^3 = 5\)[/tex]. This is the next term of the quotient.
- Multiply and subtract again: [tex]\(5(x^3 - 3) = 5x^3 - 15\)[/tex].
- Subtract: [tex]\((5x^3 - 15) - (5x^3 - 15) = 0\)[/tex].
5. Conclusion:
Since the remainder is zero, the division is exact, and the quotient is [tex]\(x + 5\)[/tex].
Therefore, the quotient of the division of [tex]\((x^4+5x^3-3x-15)\)[/tex] by [tex]\((x^3-3)\)[/tex] is:
[tex]\[
x + 5
\][/tex]
Here's how you can perform the division step-by-step:
1. Set Up the Division:
Write the dividend (the polynomial being divided) [tex]\((x^4 + 5x^3 - 3x - 15)\)[/tex] and the 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].
- [tex]\(x^4 \div x^3 = x\)[/tex]. This is the first term of the quotient.
3. Multiply and Subtract:
- Multiply [tex]\((x^3 - 3)\)[/tex] by the first term of the quotient we found, which is [tex]\(x\)[/tex].
- This gives: [tex]\(x(x^3 - 3) = x^4 - 3x\)[/tex].
- Subtract this from the original dividend: [tex]\((x^4 + 5x^3 - 3x - 15) - (x^4 - 3x) = 5x^3 - 15\)[/tex].
4. Repeat the Process:
- Divide the new leading term, [tex]\(5x^3\)[/tex], by the leading term of the divisor, [tex]\(x^3\)[/tex].
- [tex]\(5x^3 \div x^3 = 5\)[/tex]. This is the next term of the quotient.
- Multiply and subtract again: [tex]\(5(x^3 - 3) = 5x^3 - 15\)[/tex].
- Subtract: [tex]\((5x^3 - 15) - (5x^3 - 15) = 0\)[/tex].
5. Conclusion:
Since the remainder is zero, the division is exact, and the quotient is [tex]\(x + 5\)[/tex].
Therefore, the quotient of the division of [tex]\((x^4+5x^3-3x-15)\)[/tex] by [tex]\((x^3-3)\)[/tex] is:
[tex]\[
x + 5
\][/tex]
