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Dividing Polynomials

Dividing a Polynomial by a Polynomial

Example

Use long division to find (6x3 + 7x2 + 4x - 2) ÷ (2x + 1)

Solution

Step 1 Write the problem in long division form.

Algebra  
The terms of each polynomial are in descending order.  
Step 2 Divide the first term of the dividend by the first term of the divisor. Here’s a long division problem from arithmetic to help you see the similarities between the algebra and the arithmetic.
Divide 6x3 by 2x to get 3x2. Write 3x2 in the quotient line above 7x2, the x2-term of the dividend.
Step 3 Multiply the divisor by the term you found in Step 2.  
Multiply (2x + 1) by 3x2 to get 6x3 + 3x2.
Step 4 Subtract the expression you found in Step 3 from the dividend.  
Subtract (6x3 + 3x2) from (6x3 + 7x2). The result is 4x2.
Step 5 Bring down the next term from the dividend.  
Write + 4x to the right of 4x2.
Step 6 Repeat Steps 2 through 5 until the degree of the remainder is less than the degree of the divisor.  
Divide 4x2 by 2x to get 2x. Write 2x in the quotient line.

Multiply (2x + 1) by 2x to get 4x2 + 2x.

Subtract (4x2 + 2x) from (4x2 + 4x). The result is 2x.

Write -2 to the right of 2x.

Divide 2x by 2x to get 1.

Write +1 in the quotient line.

Multiply (2x + 1) by 1 to get 2x + 1.

Subtract (2x + 1) from (2x - 2).

The result is -3.

The degree of the remainder, -3, is less than the degree of the divisor, 2x + 1. So we stop.
Step 7 Write the quotient.

The quotient is:

   

Quotient is

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