%2F(x-2).jpeg "(2x^3-5x^2+3x+7)/(x-2)")
Polynomial Long Division: (2x^3-5x^2+3x+7)/(x-2)
This article will guide you through the process of dividing the polynomial 2x^3-5x^2+3x+7 by (x-2) using long division.
Understanding Polynomial Long Division
Polynomial long division is a method for dividing polynomials, similar to long division with numbers. The goal is to find the quotient and remainder of the division.
Steps for Long Division
- Set up the division: Write the dividend (2x^3-5x^2+3x+7) inside the division symbol and the divisor (x-2) outside.
- Divide the leading terms: Divide the leading term of the dividend (2x^3) by the leading term of the divisor (x), which gives 2x^2. Write this above the division symbol.
- Multiply and subtract: Multiply the quotient term (2x^2) by the divisor (x-2), which gives 2x^3 - 4x^2. Subtract this result from the first two terms of the dividend.
- Bring down the next term: Bring down the next term of the dividend (3x) to the result.
- Repeat steps 2-4: Now divide the leading term of the new dividend (x^2) by the leading term of the divisor (x), which gives x. Write this next to the 2x^2 above the division symbol.
- Continue until the degree of the remainder is less than the degree of the divisor: Continue multiplying, subtracting, and bringing down terms until the degree of the remainder is less than the degree of the divisor.
Performing the Long Division
2x^2 + x + 5
x-2 | 2x^3 - 5x^2 + 3x + 7
-(2x^3 - 4x^2)
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-x^2 + 3x
-(-x^2 + 2x)
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x + 7
-(x - 2)
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9
Result
Therefore, the result of dividing (2x^3-5x^2+3x+7) by (x-2) is:
- Quotient: 2x^2 + x + 5
- Remainder: 9
This can be written in the following form:
(2x^3-5x^2+3x+7) / (x-2) = 2x^2 + x + 5 + 9/(x-2)
Conclusion
By following the steps of polynomial long division, we successfully divided (2x^3-5x^2+3x+7) by (x-2) and obtained the quotient and remainder. This process is fundamental in understanding polynomial division and manipulating algebraic expressions.