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Answer :
To solve the problem of finding the quotient of the polynomial division of [tex]\(x^4 + 5x^3 - 3x - 15\)[/tex] by [tex]\(x^3 - 3\)[/tex], follow these steps:
1. Identify the given polynomials:
- The dividend (numerator) is [tex]\(x^4 + 5x^3 - 3x - 15\)[/tex].
- The divisor (denominator) is [tex]\(x^3 - 3\)[/tex].
2. Perform polynomial long division:
- Step 1: Divide the leading term of the numerator, [tex]\(x^4\)[/tex], by the leading term of the denominator, [tex]\(x^3\)[/tex]. This gives [tex]\(x\)[/tex].
- Step 2: Multiply the entire divisor, [tex]\(x^3 - 3\)[/tex], by [tex]\(x\)[/tex] to get [tex]\(x^4 - 3x\)[/tex].
- Step 3: Subtract [tex]\(x^4 - 3x\)[/tex] from the original dividend:
[tex]\[
(x^4 + 5x^3 - 3x - 15) - (x^4 - 3x) = 5x^3 + 0x^2 + 0x - 15
\][/tex]
- Step 4: Now divide the leading term of the result, [tex]\(5x^3\)[/tex], by the leading term of the divisor, [tex]\(x^3\)[/tex], which gives you the next term [tex]\(5\)[/tex].
- Step 5: Multiply the divisor [tex]\(x^3 - 3\)[/tex] by [tex]\(5\)[/tex], resulting in [tex]\(5x^3 - 15\)[/tex].
- Step 6: Subtract this from the current dividend:
[tex]\[
(5x^3 - 15) - (5x^3 - 15) = 0
\][/tex]
3. Conclude the division:
- Since the remainder is 0, the division is exact. The quotient obtained from the division is precisely [tex]\(x + 5\)[/tex].
Therefore, the quotient of the polynomial division is [tex]\(x + 5\)[/tex].
1. Identify the given polynomials:
- The dividend (numerator) is [tex]\(x^4 + 5x^3 - 3x - 15\)[/tex].
- The divisor (denominator) is [tex]\(x^3 - 3\)[/tex].
2. Perform polynomial long division:
- Step 1: Divide the leading term of the numerator, [tex]\(x^4\)[/tex], by the leading term of the denominator, [tex]\(x^3\)[/tex]. This gives [tex]\(x\)[/tex].
- Step 2: Multiply the entire divisor, [tex]\(x^3 - 3\)[/tex], by [tex]\(x\)[/tex] to get [tex]\(x^4 - 3x\)[/tex].
- Step 3: Subtract [tex]\(x^4 - 3x\)[/tex] from the original dividend:
[tex]\[
(x^4 + 5x^3 - 3x - 15) - (x^4 - 3x) = 5x^3 + 0x^2 + 0x - 15
\][/tex]
- Step 4: Now divide the leading term of the result, [tex]\(5x^3\)[/tex], by the leading term of the divisor, [tex]\(x^3\)[/tex], which gives you the next term [tex]\(5\)[/tex].
- Step 5: Multiply the divisor [tex]\(x^3 - 3\)[/tex] by [tex]\(5\)[/tex], resulting in [tex]\(5x^3 - 15\)[/tex].
- Step 6: Subtract this from the current dividend:
[tex]\[
(5x^3 - 15) - (5x^3 - 15) = 0
\][/tex]
3. Conclude the division:
- Since the remainder is 0, the division is exact. The quotient obtained from the division is precisely [tex]\(x + 5\)[/tex].
Therefore, the quotient of the polynomial division is [tex]\(x + 5\)[/tex].
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