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Answer :
To solve this problem, we need to find the quotient of the division of two polynomials: [tex]\(x^4 + 5x^3 - 3x - 15\)[/tex] by [tex]\(x^3 - 3\)[/tex]. We'll do this using polynomial long division.
Step 1: Set up the division
- Dividend: [tex]\(x^4 + 5x^3 - 3x - 15\)[/tex]
- Divisor: [tex]\(x^3 - 3\)[/tex]
Step 2: Divide the leading term of the dividend by the leading term of the divisor
- Divide [tex]\(x^4\)[/tex] (the leading term of the dividend) by [tex]\(x^3\)[/tex] (the leading term of the divisor). This gives us [tex]\(x\)[/tex].
Step 3: Multiply and subtract
- Multiply the entire divisor [tex]\(x^3 - 3\)[/tex] by [tex]\(x\)[/tex], which gives [tex]\(x \times (x^3 - 3) = x^4 - 3x\)[/tex].
- Subtract [tex]\(x^4 - 3x\)[/tex] from the dividend [tex]\(x^4 + 5x^3 - 3x - 15\)[/tex]:
[tex]\[
(x^4 + 5x^3 - 3x - 15) - (x^4 - 3x) = 5x^3 - 15
\][/tex]
Step 4: Repeat with the new dividend
- The new dividend is now [tex]\(5x^3 - 15\)[/tex].
Step 5: Divide the leading term of the new dividend by the leading term of the divisor
- Divide [tex]\(5x^3\)[/tex] by [tex]\(x^3\)[/tex], which gives [tex]\(5\)[/tex].
Step 6: Multiply and subtract again
- Multiply [tex]\(x^3 - 3\)[/tex] by [tex]\(5\)[/tex], which gives [tex]\(5x^3 - 15\)[/tex].
- Subtract [tex]\(5x^3 - 15\)[/tex] from the new dividend [tex]\(5x^3 - 15\)[/tex]:
[tex]\[
(5x^3 - 15) - (5x^3 - 15) = 0
\][/tex]
Result:
The division is complete and there is no remainder.
The quotient is [tex]\(x + 5\)[/tex].
Therefore, the correct answer is [tex]\(x+5\)[/tex].
Step 1: Set up the division
- Dividend: [tex]\(x^4 + 5x^3 - 3x - 15\)[/tex]
- Divisor: [tex]\(x^3 - 3\)[/tex]
Step 2: Divide the leading term of the dividend by the leading term of the divisor
- Divide [tex]\(x^4\)[/tex] (the leading term of the dividend) by [tex]\(x^3\)[/tex] (the leading term of the divisor). This gives us [tex]\(x\)[/tex].
Step 3: Multiply and subtract
- Multiply the entire divisor [tex]\(x^3 - 3\)[/tex] by [tex]\(x\)[/tex], which gives [tex]\(x \times (x^3 - 3) = x^4 - 3x\)[/tex].
- Subtract [tex]\(x^4 - 3x\)[/tex] from the dividend [tex]\(x^4 + 5x^3 - 3x - 15\)[/tex]:
[tex]\[
(x^4 + 5x^3 - 3x - 15) - (x^4 - 3x) = 5x^3 - 15
\][/tex]
Step 4: Repeat with the new dividend
- The new dividend is now [tex]\(5x^3 - 15\)[/tex].
Step 5: Divide the leading term of the new dividend by the leading term of the divisor
- Divide [tex]\(5x^3\)[/tex] by [tex]\(x^3\)[/tex], which gives [tex]\(5\)[/tex].
Step 6: Multiply and subtract again
- Multiply [tex]\(x^3 - 3\)[/tex] by [tex]\(5\)[/tex], which gives [tex]\(5x^3 - 15\)[/tex].
- Subtract [tex]\(5x^3 - 15\)[/tex] from the new dividend [tex]\(5x^3 - 15\)[/tex]:
[tex]\[
(5x^3 - 15) - (5x^3 - 15) = 0
\][/tex]
Result:
The division is complete and there is no remainder.
The quotient is [tex]\(x + 5\)[/tex].
Therefore, the correct answer is [tex]\(x+5\)[/tex].
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