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Answer :
To solve the problem of finding the quotient of [tex]\((x^4 + 5x^3 - 3x - 15)\)[/tex] divided by [tex]\((x^3 - 3)\)[/tex], we perform polynomial division.
1. Set up the division: Here, our numerator (or dividend) is [tex]\(x^4 + 5x^3 - 3x - 15\)[/tex], and our denominator (or divisor) is [tex]\(x^3 - 3\)[/tex].
2. Divide the leading terms: Look at the leading term of the dividend, [tex]\(x^4\)[/tex], and the leading term of the divisor, [tex]\(x^3\)[/tex]. Divide [tex]\(x^4\)[/tex] by [tex]\(x^3\)[/tex] to get [tex]\(x\)[/tex].
3. Multiply and subtract: Multiply the entire divisor [tex]\(x^3 - 3\)[/tex] by [tex]\(x\)[/tex] (the result from step 2), getting [tex]\(x \cdot (x^3 - 3) = x^4 - 3x\)[/tex]. Subtract this result from the original dividend:
[tex]\[
(x^4 + 5x^3 - 3x - 15) - (x^4 - 3x) = 5x^3 + 0 - 15
\][/tex]
4. Repeat the process: The new polynomial after subtraction is [tex]\(5x^3 - 15\)[/tex]. Divide the leading term, [tex]\(5x^3\)[/tex], by the leading term of the divisor, [tex]\(x^3\)[/tex], to get [tex]\(5\)[/tex].
5. Multiply and subtract again: Multiply the entire divisor [tex]\(x^3 - 3\)[/tex] by [tex]\(5\)[/tex], which results in [tex]\(5(x^3 - 3) = 5x^3 - 15\)[/tex]. Subtract this from your current polynomial:
[tex]\[
(5x^3 - 15) - (5x^3 - 15) = 0
\][/tex]
Since the remainder is zero, we have completed the division.
6. Identify the quotient: From the steps above, the quotient of the division is [tex]\(x + 5\)[/tex].
Thus, the quotient of [tex]\((x^4 + 5x^3 - 3x - 15)\)[/tex] divided by [tex]\((x^3 - 3)\)[/tex] is [tex]\(x + 5\)[/tex].
1. Set up the division: Here, our numerator (or dividend) is [tex]\(x^4 + 5x^3 - 3x - 15\)[/tex], and our denominator (or divisor) is [tex]\(x^3 - 3\)[/tex].
2. Divide the leading terms: Look at the leading term of the dividend, [tex]\(x^4\)[/tex], and the leading term of the divisor, [tex]\(x^3\)[/tex]. Divide [tex]\(x^4\)[/tex] by [tex]\(x^3\)[/tex] to get [tex]\(x\)[/tex].
3. Multiply and subtract: Multiply the entire divisor [tex]\(x^3 - 3\)[/tex] by [tex]\(x\)[/tex] (the result from step 2), getting [tex]\(x \cdot (x^3 - 3) = x^4 - 3x\)[/tex]. Subtract this result from the original dividend:
[tex]\[
(x^4 + 5x^3 - 3x - 15) - (x^4 - 3x) = 5x^3 + 0 - 15
\][/tex]
4. Repeat the process: The new polynomial after subtraction is [tex]\(5x^3 - 15\)[/tex]. Divide the leading term, [tex]\(5x^3\)[/tex], by the leading term of the divisor, [tex]\(x^3\)[/tex], to get [tex]\(5\)[/tex].
5. Multiply and subtract again: Multiply the entire divisor [tex]\(x^3 - 3\)[/tex] by [tex]\(5\)[/tex], which results in [tex]\(5(x^3 - 3) = 5x^3 - 15\)[/tex]. Subtract this from your current polynomial:
[tex]\[
(5x^3 - 15) - (5x^3 - 15) = 0
\][/tex]
Since the remainder is zero, we have completed the division.
6. Identify the quotient: From the steps above, the quotient of the division is [tex]\(x + 5\)[/tex].
Thus, the quotient of [tex]\((x^4 + 5x^3 - 3x - 15)\)[/tex] divided by [tex]\((x^3 - 3)\)[/tex] is [tex]\(x + 5\)[/tex].
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