%2F(x%5E2%2Bx%2B1).jpeg "(2x^3+3x^2+5x-4)/(x^2+x+1)")
Dividing Polynomials: (2x^3 + 3x^2 + 5x - 4) / (x^2 + x + 1)
This article will guide you through the process of dividing the polynomial 2x^3 + 3x^2 + 5x - 4 by x^2 + x + 1. We'll use the method of polynomial long division to achieve this.
Understanding Polynomial Long Division
Polynomial long division is similar to the long division you learned in arithmetic, but with polynomials instead of numbers. Here's a breakdown of the steps:
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Set up the division: Write the dividend (2x^3 + 3x^2 + 5x - 4) inside the division symbol and the divisor (x^2 + x + 1) outside.
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Divide the leading terms: Divide the leading term of the dividend (2x^3) by the leading term of the divisor (x^2). This gives us 2x. Write this result above the division symbol.
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Multiply and subtract: Multiply the divisor (x^2 + x + 1) by the result (2x). Write the product (2x^3 + 2x^2 + 2x) below the dividend. Subtract this product from the dividend.
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Bring down the next term: Bring down the next term of the dividend (-4) and add it to the result of the subtraction.
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Repeat steps 2-4: Repeat the process of dividing the leading term, multiplying, subtracting, and bringing down the next term until you reach a remainder of zero or a degree of the remainder is less than the divisor.
Performing the Division
Let's apply these steps to our problem:
2x + 1
x^2+x+1 | 2x^3 + 3x^2 + 5x - 4
-(2x^3 + 2x^2 + 2x)
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x^2 + 3x - 4
-(x^2 + x + 1)
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2x - 5
The Result
Therefore, the quotient of (2x^3 + 3x^2 + 5x - 4) divided by (x^2 + x + 1) is 2x + 1 with a remainder of 2x - 5. This can be expressed as:
(2x^3 + 3x^2 + 5x - 4) / (x^2 + x + 1) = 2x + 1 + (2x - 5) / (x^2 + x + 1)
Key Takeaways
- Polynomial long division allows you to divide polynomials efficiently.
- The steps are analogous to long division with numbers.
- The result of the division can be expressed as the quotient plus the remainder over the divisor.