Answer :
To find the quotient of the polynomial division [tex]\((x^4 + 5x^3 - 3x - 15)\)[/tex] divided by [tex]\((x^3 - 3)\)[/tex], we can use polynomial long division. Here's a detailed, step-by-step solution:
1. Set up the division: Write [tex]\((x^4 + 5x^3 - 3x - 15)\)[/tex] under the long division symbol and [tex]\((x^3 - 3)\)[/tex] outside.
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], which gives [tex]\(x \cdot (x^3 - 3) = x^4 - 3x\)[/tex]. Subtract this from the original polynomial:
[tex]\[
(x^4 + 5x^3 - 3x - 15) - (x^4 - 3x) = 5x^3 + 0 - 15
\][/tex]
4. Repeat the process: Now divide [tex]\(5x^3\)[/tex] by [tex]\(x^3\)[/tex], which gives [tex]\(5\)[/tex]. Multiply the entire divisor by [tex]\(5\)[/tex] to get [tex]\(5(x^3 - 3) = 5x^3 - 15\)[/tex].
5. Subtract again:
[tex]\[
(5x^3 + 0 - 15) - (5x^3 - 15) = 0
\][/tex]
Since we have reached a remainder of [tex]\(0\)[/tex], the division is complete. The quotient of the division is the sum of the individual quotients obtained in the steps:
The quotient is [tex]\(x + 5\)[/tex].
Therefore, the answer is [tex]\(x + 5\)[/tex].
1. Set up the division: Write [tex]\((x^4 + 5x^3 - 3x - 15)\)[/tex] under the long division symbol and [tex]\((x^3 - 3)\)[/tex] outside.
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], which gives [tex]\(x \cdot (x^3 - 3) = x^4 - 3x\)[/tex]. Subtract this from the original polynomial:
[tex]\[
(x^4 + 5x^3 - 3x - 15) - (x^4 - 3x) = 5x^3 + 0 - 15
\][/tex]
4. Repeat the process: Now divide [tex]\(5x^3\)[/tex] by [tex]\(x^3\)[/tex], which gives [tex]\(5\)[/tex]. Multiply the entire divisor by [tex]\(5\)[/tex] to get [tex]\(5(x^3 - 3) = 5x^3 - 15\)[/tex].
5. Subtract again:
[tex]\[
(5x^3 + 0 - 15) - (5x^3 - 15) = 0
\][/tex]
Since we have reached a remainder of [tex]\(0\)[/tex], the division is complete. The quotient of the division is the sum of the individual quotients obtained in the steps:
The quotient is [tex]\(x + 5\)[/tex].
Therefore, the answer is [tex]\(x + 5\)[/tex].
