%2F(x%2B2)-long-division.jpeg "(x^3-4x^2+3x+2)/(x+2) Long Division")
Long Division of Polynomials: (x^3 - 4x^2 + 3x + 2) / (x + 2)
This article will guide you through the process of dividing the polynomial x^3 - 4x^2 + 3x + 2 by the binomial x + 2 using long division.
Understanding Long Division with Polynomials
Long division with polynomials works similarly to long division with numbers. We follow these steps:
- Set up the division: Arrange the dividend (x^3 - 4x^2 + 3x + 2) and divisor (x + 2) in the traditional long division format.
- Divide the leading terms: Divide the leading term of the dividend (x^3) by the leading term of the divisor (x). This gives us x^2.
- Multiply and subtract: Multiply the quotient (x^2) by the divisor (x + 2), resulting in x^3 + 2x^2. Subtract this product from the dividend.
- Bring down the next term: Bring down the next term from the dividend (3x).
- Repeat steps 2-4: Repeat the process of dividing, multiplying, and subtracting until you reach a remainder that has a lower degree than the divisor.
Applying Long Division
Let's perform the long division for (x^3 - 4x^2 + 3x + 2) / (x + 2):
x^2 - 6x + 15
x + 2 | x^3 - 4x^2 + 3x + 2
x^3 + 2x^2
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-6x^2 + 3x
-6x^2 - 12x
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15x + 2
15x + 30
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-28
Interpretation of the Result
The result of the long division is:
- Quotient: x^2 - 6x + 15
- Remainder: -28
This means that:
(x^3 - 4x^2 + 3x + 2) / (x + 2) = x^2 - 6x + 15 - 28/(x + 2)
Conclusion
Long division of polynomials is a powerful tool for dividing polynomials and expressing the result in a more manageable form. This method can be applied to various polynomial expressions and helps us understand the relationship between the dividend, divisor, quotient, and remainder.