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 use polynomial long division. Let's break down the steps:
1. Setup the Division: We need to divide the polynomial [tex]\(x^4 + 5x^3 - 3x - 15\)[/tex] by [tex]\(x^3 - 3\)[/tex].
2. Divide the First Term: Divide the leading term of the dividend [tex]\(x^4\)[/tex] by the leading term of the divisor [tex]\(x^3\)[/tex] to get the first term of the quotient:
[tex]\[
\frac{x^4}{x^3} = x
\][/tex]
3. Multiply and Subtract: Multiply the entire divisor [tex]\((x^3 - 3)\)[/tex] by this first term [tex]\(x\)[/tex] and subtract the result from the original polynomial:
[tex]\[
(x^3 - 3) \times x = x^4 - 3x
\][/tex]
[tex]\[
(x^4 + 5x^3 - 3x - 15) - (x^4 - 3x) = 5x^3 + 0x^2 - (3x - (-3x)) - 15
\][/tex]
4. Result: After subtracting, the new polynomial becomes:
[tex]\[
5x^3 - 15
\][/tex]
5. Repeat Process: Divide the next leading term [tex]\(5x^3\)[/tex] by [tex]\(x^3\)[/tex], giving [tex]\(5\)[/tex]. Multiply the divisor by [tex]\(5\)[/tex] and subtract once more:
[tex]\[
(x^3 - 3) \times 5 = 5x^3 - 15
\][/tex]
[tex]\[
(5x^3 - 15) - (5x^3 - 15) = 0
\][/tex]
6. Conclusion: After subtracting, the remainder is [tex]\(0\)[/tex], which means the division ends here.
Thus, the quotient obtained from the division of [tex]\((x^4 + 5x^3 - 3x - 15)\)[/tex] by [tex]\((x^3 - 3)\)[/tex] is [tex]\(x + 5\)[/tex]. Therefore, the correct answer to the question is:
[tex]\[ x + 5 \][/tex]
1. Setup the Division: We need to divide the polynomial [tex]\(x^4 + 5x^3 - 3x - 15\)[/tex] by [tex]\(x^3 - 3\)[/tex].
2. Divide the First Term: Divide the leading term of the dividend [tex]\(x^4\)[/tex] by the leading term of the divisor [tex]\(x^3\)[/tex] to get the first term of the quotient:
[tex]\[
\frac{x^4}{x^3} = x
\][/tex]
3. Multiply and Subtract: Multiply the entire divisor [tex]\((x^3 - 3)\)[/tex] by this first term [tex]\(x\)[/tex] and subtract the result from the original polynomial:
[tex]\[
(x^3 - 3) \times x = x^4 - 3x
\][/tex]
[tex]\[
(x^4 + 5x^3 - 3x - 15) - (x^4 - 3x) = 5x^3 + 0x^2 - (3x - (-3x)) - 15
\][/tex]
4. Result: After subtracting, the new polynomial becomes:
[tex]\[
5x^3 - 15
\][/tex]
5. Repeat Process: Divide the next leading term [tex]\(5x^3\)[/tex] by [tex]\(x^3\)[/tex], giving [tex]\(5\)[/tex]. Multiply the divisor by [tex]\(5\)[/tex] and subtract once more:
[tex]\[
(x^3 - 3) \times 5 = 5x^3 - 15
\][/tex]
[tex]\[
(5x^3 - 15) - (5x^3 - 15) = 0
\][/tex]
6. Conclusion: After subtracting, the remainder is [tex]\(0\)[/tex], which means the division ends here.
Thus, the quotient obtained from the division of [tex]\((x^4 + 5x^3 - 3x - 15)\)[/tex] by [tex]\((x^3 - 3)\)[/tex] is [tex]\(x + 5\)[/tex]. Therefore, the correct answer to the question is:
[tex]\[ x + 5 \][/tex]
