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Answer :
To solve the problem of finding the quotient of the polynomials [tex]\((x^4 + 5x^3 - 3x - 15)\)[/tex] divided by [tex]\((x^3 - 3)\)[/tex], we use polynomial division. We'll break it down step-by-step:
1. Set Up the Division:
Start by writing the dividend [tex]\(x^4 + 5x^3 - 3x - 15\)[/tex] and the divisor [tex]\(x^3 - 3\)[/tex].
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]:
[tex]\[
\frac{x^4}{x^3} = x
\][/tex]
This is the first term of the quotient.
3. Multiply and Subtract:
Multiply [tex]\(x\)[/tex] by the divisor [tex]\(x^3 - 3\)[/tex]:
[tex]\[
x \cdot (x^3 - 3) = x^4 - 3x
\][/tex]
Subtract this result from the original dividend:
[tex]\[
(x^4 + 5x^3 - 3x - 15) - (x^4 - 3x) = 5x^3 + 0x - 15
\][/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]:
[tex]\[
\frac{5x^3}{x^3} = 5
\][/tex]
This is the next term of the quotient.
- Multiply [tex]\(5\)[/tex] by the divisor [tex]\(x^3 - 3\)[/tex]:
[tex]\[
5 \cdot (x^3 - 3) = 5x^3 - 15
\][/tex]
- Subtract this result:
[tex]\[
(5x^3 - 15) - (5x^3 - 15) = 0
\][/tex]
5. Conclude the Division:
Since the remainder is 0, the division is complete, and our quotient is [tex]\((x + 5)\)[/tex].
Therefore, the quotient of the division is [tex]\(x + 5\)[/tex].
1. Set Up the Division:
Start by writing the dividend [tex]\(x^4 + 5x^3 - 3x - 15\)[/tex] and the divisor [tex]\(x^3 - 3\)[/tex].
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]:
[tex]\[
\frac{x^4}{x^3} = x
\][/tex]
This is the first term of the quotient.
3. Multiply and Subtract:
Multiply [tex]\(x\)[/tex] by the divisor [tex]\(x^3 - 3\)[/tex]:
[tex]\[
x \cdot (x^3 - 3) = x^4 - 3x
\][/tex]
Subtract this result from the original dividend:
[tex]\[
(x^4 + 5x^3 - 3x - 15) - (x^4 - 3x) = 5x^3 + 0x - 15
\][/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]:
[tex]\[
\frac{5x^3}{x^3} = 5
\][/tex]
This is the next term of the quotient.
- Multiply [tex]\(5\)[/tex] by the divisor [tex]\(x^3 - 3\)[/tex]:
[tex]\[
5 \cdot (x^3 - 3) = 5x^3 - 15
\][/tex]
- Subtract this result:
[tex]\[
(5x^3 - 15) - (5x^3 - 15) = 0
\][/tex]
5. Conclude the Division:
Since the remainder is 0, the division is complete, and our quotient is [tex]\((x + 5)\)[/tex].
Therefore, the quotient of the division is [tex]\(x + 5\)[/tex].
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