(2x^3+7x^2+2x+9)/(2x+3)

Jasonbradley

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Jun 16, 2024

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Polynomial Division: (2x^3+7x^2+2x+9)/(2x+3)

This article will walk you through the process of dividing the polynomial (2x^3+7x^2+2x+9) by the binomial (2x+3) using polynomial long division.

Understanding Polynomial Long Division

Polynomial long division is a method for dividing polynomials, similar to long division with numbers. Here's how it works:

1. Set up the division: Write the dividend (the polynomial being divided) inside the division symbol, and the divisor (the polynomial dividing) outside.
2. Divide the leading terms: Focus on the leading terms of the dividend and divisor. Divide the leading term of the dividend by the leading term of the divisor.
3. Multiply and subtract: Multiply the result from step 2 by the entire divisor. Subtract this product from the dividend.
4. Bring down the next term: Bring down the next term of the dividend.
5. Repeat steps 2-4: Repeat the process of dividing, multiplying, subtracting, and bringing down until you reach a remainder that has a lower degree than the divisor.

Example: (2x^3+7x^2+2x+9)/(2x+3)

Let's divide (2x^3+7x^2+2x+9) by (2x+3) using polynomial long division:

        x^2 + 2x - 1       
    _____________________
2x+3 | 2x^3 + 7x^2 + 2x + 9 
       -(2x^3 + 3x^2)       
       _____________________
              4x^2 + 2x
              -(4x^2 + 6x)
              _____________________
                      -4x + 9
                      -(-4x - 6)
                      _____________________
                              15

Explanation:

1. Divide leading terms: (2x^3) / (2x) = x^2. Write x^2 above the division symbol.
2. Multiply and subtract: (x^2) * (2x+3) = 2x^3 + 3x^2. Subtract this from the dividend.
3. Bring down: Bring down the next term, which is 2x.
4. Repeat steps 2-4: (4x^2) / (2x) = 2x. Multiply and subtract, then bring down the next term (9). Continue until you reach a remainder of 15.

The Result

The quotient of (2x^3+7x^2+2x+9) divided by (2x+3) is x^2 + 2x - 1, with a remainder of 15. We can express this in the following form:

(2x^3+7x^2+2x+9) / (2x+3) = x^2 + 2x - 1 + 15/(2x+3)

Key Takeaways

Polynomial long division can be a helpful tool for simplifying rational expressions or finding factors of polynomials. Remember to carefully follow the steps and pay close attention to signs and coefficients throughout the process.