(2a2 + A + 3) ÷ (a - 1)

Jasonbradley

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Jun 16, 2024

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Dividing Polynomials: (2a² + a + 3) ÷ (a - 1)

In this article, we will explore the process of dividing the polynomial (2a² + a + 3) by (a - 1). We'll use the method of long division to achieve this.

Long Division of Polynomials

Long division for polynomials follows a similar procedure to long division for numbers. Here's how to apply it to our problem:

1. Set up the division:

        _______
a - 1 | 2a² + a + 3 

2. Divide the leading terms:

• Divide the leading term of the dividend (2a²) by the leading term of the divisor (a). This gives us 2a.

• Write 2a above the line as the first term of the quotient.

        2a _______
a - 1 | 2a² + a + 3 

3. Multiply the divisor by the quotient term:

• Multiply (a - 1) by 2a, which gives 2a² - 2a.

• Write this result below the dividend.

        2a _______
a - 1 | 2a² + a + 3 
        2a² - 2a

4. Subtract:

• Subtract the result from the previous step from the dividend. Remember to change the signs when subtracting.

        2a _______
a - 1 | 2a² + a + 3 
        2a² - 2a
         -------
              3a + 3 

5. Bring down the next term:

• Bring down the next term from the dividend (+3).

        2a _______
a - 1 | 2a² + a + 3 
        2a² - 2a
         -------
              3a + 3 

6. Repeat steps 2-5:

• Divide the new leading term (3a) by the leading term of the divisor (a), which gives 3.

• Write 3 next to 2a in the quotient.

• Multiply (a - 1) by 3 to get 3a - 3.

• Subtract this result from the previous line.

        2a + 3 _______
a - 1 | 2a² + a + 3 
        2a² - 2a
         -------
              3a + 3 
              3a - 3 
              -----
                  6

7. The Remainder:

• The final result is 6. This is the remainder.

Solution

Therefore, the division of (2a² + a + 3) by (a - 1) can be expressed as:

(2a² + a + 3) ÷ (a - 1) = 2a + 3 + 6/(a - 1)

This means that the quotient is 2a + 3 with a remainder of 6.