Answer :
To determine the quotient of the polynomials [tex]\((x^4 + 5x^3 - 3x - 15)\)[/tex] and [tex]\((x^3 - 3)\)[/tex], we perform polynomial division. Here is a step-by-step explanation of how you would do this division:
1. Set up the problem:
Write the dividend [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]). This gives [tex]\(x\)[/tex].
3. Multiply and subtract:
Multiply the entire divisor [tex]\(x^3 - 3\)[/tex] by the result ([tex]\(x\)[/tex]) to get [tex]\(x \cdot (x^3 - 3) = x^4 - 3x\)[/tex].
4. Subtract the result from the dividend:
[tex]\((x^4 + 5x^3 - 3x - 15) - (x^4 - 3x) = 5x^3 + 0x^2 + 0x - 15\)[/tex].
5. Repeat the process:
- Divide the new leading term [tex]\(5x^3\)[/tex] by [tex]\(x^3\)[/tex] to get [tex]\(5\)[/tex].
- Multiply the divisor by [tex]\(5\)[/tex] to get [tex]\(5(x^3 - 3) = 5x^3 - 15\)[/tex].
- Subtract this product from the remaining terms:
[tex]\((5x^3 + 0x^2 + 0x - 15) - (5x^3 - 15) = 0\)[/tex].
6. Result of the division:
The quotient is [tex]\(x + 5\)[/tex].
By following these steps, the quotient obtained from dividing [tex]\((x^4 + 5x^3 - 3x - 15)\)[/tex] by [tex]\((x^3 - 3)\)[/tex] is indeed:
[tex]\[ x + 5 \][/tex]
1. Set up the problem:
Write the dividend [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]). This gives [tex]\(x\)[/tex].
3. Multiply and subtract:
Multiply the entire divisor [tex]\(x^3 - 3\)[/tex] by the result ([tex]\(x\)[/tex]) to get [tex]\(x \cdot (x^3 - 3) = x^4 - 3x\)[/tex].
4. Subtract the result from the dividend:
[tex]\((x^4 + 5x^3 - 3x - 15) - (x^4 - 3x) = 5x^3 + 0x^2 + 0x - 15\)[/tex].
5. Repeat the process:
- Divide the new leading term [tex]\(5x^3\)[/tex] by [tex]\(x^3\)[/tex] to get [tex]\(5\)[/tex].
- Multiply the divisor by [tex]\(5\)[/tex] to get [tex]\(5(x^3 - 3) = 5x^3 - 15\)[/tex].
- Subtract this product from the remaining terms:
[tex]\((5x^3 + 0x^2 + 0x - 15) - (5x^3 - 15) = 0\)[/tex].
6. Result of the division:
The quotient is [tex]\(x + 5\)[/tex].
By following these steps, the quotient obtained from dividing [tex]\((x^4 + 5x^3 - 3x - 15)\)[/tex] by [tex]\((x^3 - 3)\)[/tex] is indeed:
[tex]\[ x + 5 \][/tex]
