(2x^3-5x-7)/(x-2) Quizizz

Jasonbradley

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Jun 16, 2024

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Dividing Polynomials: (2x³ - 5x - 7) / (x - 2)

This article will guide you through the process of dividing the polynomial (2x³ - 5x - 7) by (x - 2). This is a common task encountered in algebra, particularly when working with rational expressions. Let's break down the steps and understand the concept behind polynomial division.

Understanding Polynomial Division

Polynomial division is similar to long division with numbers. It involves finding the quotient and remainder when one polynomial is divided by another.

Step-by-Step Process:

1. Set up the division:

   • Write the dividend (2x³ - 5x - 7) inside the division symbol.
   • Write the divisor (x - 2) outside the division symbol.

        ________
   x - 2 | 2x³ - 5x - 7 
   

2. Divide the leading terms:

   • Divide the leading term of the dividend (2x³) by the leading term of the divisor (x).
   • Write the result (2x²) above the division symbol.

        2x² ______
   x - 2 | 2x³ - 5x - 7 
   

3. Multiply the quotient by the divisor:

   • Multiply the quotient (2x²) by the entire divisor (x - 2).
   • Write the result (2x³ - 4x²) below the dividend.

        2x² ______
   x - 2 | 2x³ - 5x - 7 
           2x³ - 4x²
   

4. Subtract:

   • Subtract the result from the dividend. Remember to change the signs of the terms in the bottom row.

        2x² ______
   x - 2 | 2x³ - 5x - 7 
           2x³ - 4x²
           -------
                 x - 7 
   

5. Bring down the next term:

   • Bring down the next term of the dividend (-7).

        2x² ______
   x - 2 | 2x³ - 5x - 7 
           2x³ - 4x²
           -------
                 x - 7 
   

6. Repeat steps 2-5:

   • Divide the new leading term (x) by the divisor's leading term (x).
   • Write the result (1) above the division symbol.
   • Multiply the quotient (1) by the divisor (x - 2) and write the result below.
   • Subtract.

        2x² + 1 ___
   x - 2 | 2x³ - 5x - 7 
           2x³ - 4x²
           -------
                 x - 7 
                 x - 2 
   

7. Final subtraction:

   • Subtract the last result from the previous line.

        2x² + 1 ___
   x - 2 | 2x³ - 5x - 7 
           2x³ - 4x²
           -------
                 x - 7 
                 x - 2 
           -------
                 -5
   

8. The remainder:

   • The final result (-5) is the remainder.

Conclusion:

The result of dividing (2x³ - 5x - 7) by (x - 2) is:

(2x³ - 5x - 7) / (x - 2) = 2x² + 1 - 5/(x - 2)

You can express this result as a mixed number with a polynomial quotient and a rational remainder. This process helps to understand the relationship between polynomials and their factors and is a crucial concept in algebra.

Remember, practice makes perfect! Continue to work through examples of polynomial division to solidify your understanding.