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Answer :
To find the quotient of the polynomial division [tex]\((x^4 + 5x^3 - 3x - 15)\)[/tex] divided by [tex]\((x^3 - 3)\)[/tex], you can use polynomial long division. Here’s a step-by-step explanation:
1. Setup the Division:
We are dividing the polynomial [tex]\(x^4 + 5x^3 - 3x - 15\)[/tex] by [tex]\(x^3 - 3\)[/tex].
2. Divide the leading terms:
Look at the leading term of the dividend [tex]\(x^4\)[/tex] and divide it by the leading term of the divisor [tex]\(x^3\)[/tex]. This gives us [tex]\(x^1\)[/tex] or just [tex]\(x\)[/tex].
3. Multiply and Subtract:
Multiply the entire divisor [tex]\(x^3 - 3\)[/tex] by the result from the previous step [tex]\(x\)[/tex]. This gives you [tex]\(x(x^3 - 3) = x^4 - 3x\)[/tex].
Subtract this result from the original polynomial:
[tex]\[(x^4 + 5x^3 - 3x - 15) - (x^4 - 3x) = 5x^3 - 0x - 15\][/tex]
4. Repeat the process:
Divide the new 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 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 result of the previous subtraction:
[tex]\[(5x^3 - 0x - 15) - (5x^3 - 15) = 0\][/tex]
6. Conclusion:
Since the remainder is [tex]\(0\)[/tex], the original polynomial is exactly divisible, and the quotient is [tex]\(x + 5\)[/tex].
Therefore, the quotient of [tex]\((x^4 + 5x^3 - 3x - 15)\)[/tex] divided by [tex]\((x^3 - 3)\)[/tex] is [tex]\(x + 5\)[/tex].
1. Setup the Division:
We are dividing the polynomial [tex]\(x^4 + 5x^3 - 3x - 15\)[/tex] by [tex]\(x^3 - 3\)[/tex].
2. Divide the leading terms:
Look at the leading term of the dividend [tex]\(x^4\)[/tex] and divide it by the leading term of the divisor [tex]\(x^3\)[/tex]. This gives us [tex]\(x^1\)[/tex] or just [tex]\(x\)[/tex].
3. Multiply and Subtract:
Multiply the entire divisor [tex]\(x^3 - 3\)[/tex] by the result from the previous step [tex]\(x\)[/tex]. This gives you [tex]\(x(x^3 - 3) = x^4 - 3x\)[/tex].
Subtract this result from the original polynomial:
[tex]\[(x^4 + 5x^3 - 3x - 15) - (x^4 - 3x) = 5x^3 - 0x - 15\][/tex]
4. Repeat the process:
Divide the new 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 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 result of the previous subtraction:
[tex]\[(5x^3 - 0x - 15) - (5x^3 - 15) = 0\][/tex]
6. Conclusion:
Since the remainder is [tex]\(0\)[/tex], the original polynomial is exactly divisible, and the quotient is [tex]\(x + 5\)[/tex].
Therefore, 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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