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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 will perform polynomial division.
1. Set up the polynomial division:
We are dividing:
- Dividend: [tex]\(x^4 + 5x^3 - 3x - 15\)[/tex]
- Divisor: [tex]\(x^3 - 3\)[/tex]
2. Perform the division:
- Step 1: Divide the leading term of the dividend ([tex]\(x^4\)[/tex]) by the leading term of the divisor ([tex]\(x^3\)[/tex]). This gives us [tex]\(x\)[/tex].
- Step 2: Multiply the entire divisor [tex]\(x^3 - 3\)[/tex] by this result [tex]\(x\)[/tex], which gives [tex]\(x \cdot (x^3 - 3) = x^4 - 3x\)[/tex].
- Step 3: Subtract this result from the original dividend:
[tex]\[
(x^4 + 5x^3 - 3x - 15) - (x^4 - 3x) = 5x^3 + 0x^2 + 0x - 15
\][/tex]
- Step 4: Now take the next leading term from the result ([tex]\(5x^3\)[/tex]) and divide by [tex]\(x^3\)[/tex], which gives [tex]\(5\)[/tex].
- Step 5: Multiply the entire divisor [tex]\(x^3 - 3\)[/tex] by this new result [tex]\(5\)[/tex], which gives us [tex]\(5 \cdot (x^3 - 3) = 5x^3 - 15\)[/tex].
- Step 6: Subtract this from the current polynomial:
[tex]\[
(5x^3 - 15) - (5x^3 - 15) = 0
\][/tex]
3. Conclusion:
After performing the polynomial division, we find that the quotient is [tex]\(x + 5\)[/tex] and a remainder of 0. This means the original numerator divides evenly by the divisor.
Thus, the quotient of [tex]\((x^4 + 5x^3 - 3x - 15)\)[/tex] divided by [tex]\((x^3 - 3)\)[/tex] is [tex]\(\boxed{x + 5}\)[/tex].
1. Set up the polynomial division:
We are dividing:
- Dividend: [tex]\(x^4 + 5x^3 - 3x - 15\)[/tex]
- Divisor: [tex]\(x^3 - 3\)[/tex]
2. Perform the division:
- Step 1: Divide the leading term of the dividend ([tex]\(x^4\)[/tex]) by the leading term of the divisor ([tex]\(x^3\)[/tex]). This gives us [tex]\(x\)[/tex].
- Step 2: Multiply the entire divisor [tex]\(x^3 - 3\)[/tex] by this result [tex]\(x\)[/tex], which gives [tex]\(x \cdot (x^3 - 3) = x^4 - 3x\)[/tex].
- Step 3: Subtract this result from the original dividend:
[tex]\[
(x^4 + 5x^3 - 3x - 15) - (x^4 - 3x) = 5x^3 + 0x^2 + 0x - 15
\][/tex]
- Step 4: Now take the next leading term from the result ([tex]\(5x^3\)[/tex]) and divide by [tex]\(x^3\)[/tex], which gives [tex]\(5\)[/tex].
- Step 5: Multiply the entire divisor [tex]\(x^3 - 3\)[/tex] by this new result [tex]\(5\)[/tex], which gives us [tex]\(5 \cdot (x^3 - 3) = 5x^3 - 15\)[/tex].
- Step 6: Subtract this from the current polynomial:
[tex]\[
(5x^3 - 15) - (5x^3 - 15) = 0
\][/tex]
3. Conclusion:
After performing the polynomial division, we find that the quotient is [tex]\(x + 5\)[/tex] and a remainder of 0. This means the original numerator divides evenly by the divisor.
Thus, the quotient of [tex]\((x^4 + 5x^3 - 3x - 15)\)[/tex] divided by [tex]\((x^3 - 3)\)[/tex] is [tex]\(\boxed{x + 5}\)[/tex].
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