Polynomial Long Division: (x^3+3x^2-6x+2)/(x-1)
This article explores the process of dividing the polynomial x^3 + 3x^2 - 6x + 2 by x - 1 using polynomial long division.
Understanding Polynomial Long Division
Polynomial long division is a method for dividing polynomials, similar to the long division method used for numbers. It helps us to rewrite a polynomial as a product of two polynomials (quotient and divisor) plus a remainder.
Steps for Polynomial Long Division
Let's break down the division of (x^3 + 3x^2 - 6x + 2) by (x - 1):
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Set up the division:
________ x - 1 | x^3 + 3x^2 - 6x + 2
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Divide the leading terms:
- The leading term of the dividend (x^3) is divided by the leading term of the divisor (x).
- The result is x^2, which is placed above the x^2 term in the quotient.
x^2 ______ x - 1 | x^3 + 3x^2 - 6x + 2
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Multiply the divisor by the quotient term:
- Multiply (x - 1) by x^2, which gives x^3 - x^2.
x^2 ______ x - 1 | x^3 + 3x^2 - 6x + 2 x^3 - x^2
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Subtract:
- Subtract the product obtained in the previous step from the dividend.
- Change the signs of the terms in the product and add.
x^2 ______ x - 1 | x^3 + 3x^2 - 6x + 2 x^3 - x^2 ------- 4x^2
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Bring down the next term:
- Bring down the next term of the dividend (-6x).
x^2 ______ x - 1 | x^3 + 3x^2 - 6x + 2 x^3 - x^2 ------- 4x^2 - 6x
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Repeat steps 2-5:
- Divide the new leading term (4x^2) by the leading term of the divisor (x), which gives 4x.
- Multiply (x - 1) by 4x to get 4x^2 - 4x.
- Subtract the product, changing signs and adding.
- Bring down the next term (+2).
x^2 + 4x ______ x - 1 | x^3 + 3x^2 - 6x + 2 x^3 - x^2 ------- 4x^2 - 6x 4x^2 - 4x ------- -2x + 2
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Final step:
- Divide -2x by x, which gives -2.
- Multiply (x - 1) by -2 to get -2x + 2.
- Subtract, changing signs and adding.
x^2 + 4x - 2 _____ x - 1 | x^3 + 3x^2 - 6x + 2 x^3 - x^2 ------- 4x^2 - 6x 4x^2 - 4x ------- -2x + 2 -2x + 2 ------- 0
Result
Therefore, the division of (x^3 + 3x^2 - 6x + 2) by (x - 1) gives us:
Quotient: x^2 + 4x - 2 Remainder: 0
This can be expressed as:
(x^3 + 3x^2 - 6x + 2) / (x - 1) = x^2 + 4x - 2
Conclusion
Polynomial long division allows us to systematically divide polynomials and express the result as a quotient and a remainder. This method is crucial for simplifying expressions, factoring polynomials, and solving algebraic equations.