Answer :
To find the quotient of the polynomial division of [tex]\((x^4 + 5x^3 - 3x - 15)\)[/tex] by [tex]\((x^3 - 3)\)[/tex], we perform polynomial long division:
1. Divide the First Terms: Start by dividing the highest degree term of the numerator, [tex]\(x^4\)[/tex], by the highest degree term of the denominator, [tex]\(x^3\)[/tex]. Doing this gives you [tex]\(x\)[/tex].
2. Multiply and Subtract: Multiply the entire divisor [tex]\((x^3 - 3)\)[/tex] by the term you found, [tex]\(x\)[/tex], which gives [tex]\(x(x^3 - 3) = x^4 - 3x\)[/tex]. Subtract this from the original polynomial:
[tex]\[
(x^4 + 5x^3 - 3x - 15) - (x^4 - 3x) = 5x^3 + 0x^2 - 3x - 15.
\][/tex]
3. Repeat the Process: Now, divide the new highest degree term, [tex]\(5x^3\)[/tex], by the highest term of the divisor, [tex]\(x^3\)[/tex], which is [tex]\(5\)[/tex].
4. Multiply and Subtract Again: Multiply the divisor by [tex]\(5\)[/tex], giving [tex]\(5(x^3 - 3) = 5x^3 - 15\)[/tex]. Subtract this from the polynomial:
[tex]\[
(5x^3 - 3x - 15) - (5x^3 - 15) = 0x^2 - 3x + 0.
\][/tex]
5. Conclusion: There are no more terms in the quotient to find, so the quotient of the division is [tex]\(x + 5\)[/tex]. The remainder is zero, which confirms that the division results in a polynomial without any remaining part.
Therefore, the quotient of the given division is [tex]\(\boxed{x + 5}\)[/tex].
1. Divide the First Terms: Start by dividing the highest degree term of the numerator, [tex]\(x^4\)[/tex], by the highest degree term of the denominator, [tex]\(x^3\)[/tex]. Doing this gives you [tex]\(x\)[/tex].
2. Multiply and Subtract: Multiply the entire divisor [tex]\((x^3 - 3)\)[/tex] by the term you found, [tex]\(x\)[/tex], which gives [tex]\(x(x^3 - 3) = x^4 - 3x\)[/tex]. Subtract this from the original polynomial:
[tex]\[
(x^4 + 5x^3 - 3x - 15) - (x^4 - 3x) = 5x^3 + 0x^2 - 3x - 15.
\][/tex]
3. Repeat the Process: Now, divide the new highest degree term, [tex]\(5x^3\)[/tex], by the highest term of the divisor, [tex]\(x^3\)[/tex], which is [tex]\(5\)[/tex].
4. Multiply and Subtract Again: Multiply the divisor by [tex]\(5\)[/tex], giving [tex]\(5(x^3 - 3) = 5x^3 - 15\)[/tex]. Subtract this from the polynomial:
[tex]\[
(5x^3 - 3x - 15) - (5x^3 - 15) = 0x^2 - 3x + 0.
\][/tex]
5. Conclusion: There are no more terms in the quotient to find, so the quotient of the division is [tex]\(x + 5\)[/tex]. The remainder is zero, which confirms that the division results in a polynomial without any remaining part.
Therefore, the quotient of the given division is [tex]\(\boxed{x + 5}\)[/tex].
