Answer :
To find the quotient of the polynomial division [tex]\((x^4 + 5x^3 - 3x - 15) \div (x^3 - 3)\)[/tex], follow these steps:
1. Set up the division: Divide the first term of the numerator, [tex]\(x^4\)[/tex], by the first term of the denominator, [tex]\(x^3\)[/tex]. This gives [tex]\(x\)[/tex].
2. Multiply and subtract: Multiply the entire divisor [tex]\((x^3 - 3)\)[/tex] by [tex]\(x\)[/tex] to get [tex]\(x(x^3 - 3) = x^4 - 3x\)[/tex]. Subtract this from the original polynomial:
[tex]\[
(x^4 + 5x^3 - 3x - 15) - (x^4 - 3x) = 5x^3 - 0x^2 + 0x - 15
\][/tex]
3. Repeat the process: Divide the new leading term [tex]\(5x^3\)[/tex] by [tex]\(x^3\)[/tex], which is [tex]\(5\)[/tex].
4. Multiply and subtract again: Multiply the entire divisor [tex]\((x^3 - 3)\)[/tex] by [tex]\(5\)[/tex] to get [tex]\(5(x^3 - 3) = 5x^3 - 15\)[/tex]. Subtract this from the current polynomial:
[tex]\[
(5x^3 - 15) - (5x^3 - 15) = 0
\][/tex]
5. Result: The division has no remainder, and the quotient is [tex]\(x + 5\)[/tex].
Thus, the quotient of the division is [tex]\(x + 5\)[/tex].
1. Set up the division: Divide the first term of the numerator, [tex]\(x^4\)[/tex], by the first term of the denominator, [tex]\(x^3\)[/tex]. This gives [tex]\(x\)[/tex].
2. Multiply and subtract: Multiply the entire divisor [tex]\((x^3 - 3)\)[/tex] by [tex]\(x\)[/tex] to get [tex]\(x(x^3 - 3) = x^4 - 3x\)[/tex]. Subtract this from the original polynomial:
[tex]\[
(x^4 + 5x^3 - 3x - 15) - (x^4 - 3x) = 5x^3 - 0x^2 + 0x - 15
\][/tex]
3. Repeat the process: Divide the new leading term [tex]\(5x^3\)[/tex] by [tex]\(x^3\)[/tex], which is [tex]\(5\)[/tex].
4. Multiply and subtract again: Multiply the entire divisor [tex]\((x^3 - 3)\)[/tex] by [tex]\(5\)[/tex] to get [tex]\(5(x^3 - 3) = 5x^3 - 15\)[/tex]. Subtract this from the current polynomial:
[tex]\[
(5x^3 - 15) - (5x^3 - 15) = 0
\][/tex]
5. Result: The division has no remainder, and the quotient is [tex]\(x + 5\)[/tex].
Thus, the quotient of the division is [tex]\(x + 5\)[/tex].
