10. Dividing one polynomial by another

The lesson, step by step

1. Warm-up

   Multiply 2a − 3x by 3a − 2x. (6a² − 13ax + 6x².) Keep the answer; you will divide it back in a moment.

2. The rule

   Read Art. 79 and its rule:

   Arrange the dividend and divisor with reference to the leading letter, and place the divisor on the right of the dividend.

   Divide the first term of the dividend by the first term of the divisor, for the first term of the quotient.

   SayThen multiply the whole divisor by that term, subtract, and repeat with what is left.

3. Work the book's first example together

   1. Divide 6a²—13ax+6x² by 2a— 3x.

   6a² ÷ 2a = 3a. Multiply: 3a(2a − 3x) = 6a² − 9ax. Subtract: −4ax + 6x².

   −4ax ÷ 2a = −2x. Multiply: −2x(2a − 3x) = −4ax + 6x². Subtract: 0.

   Quotient: 3a − 2x, exactly the warm-up factor.

4. Look at examples 2 to 5

   Read the other worked examples. Point out example 3: a³ + x³ divided by a + x, where the missing powers of a get room left for them in the work.

5. Guided practice

   Cover the answers. Your student does these while you watch:

   7. Divide 2x²+7xy+6y² by x+2y.

   11. Divide ac+bc — ad — bd by a+b. . . Ans. c — d.

   (The first comes out 2x + 3y.)

6. Independent practice

   Exercises 6, 8, 9 and 10. Check each by multiplying the quotient by the divisor.

7. Check

   After working one by hand, your student can confirm it on the site's calculator by expanding quotient times divisor, for example expand((x+2*y)*(2*x+3*y)).

8. Wrap-up

   Ask: How do you know when to stop? (When the remainder is 0, or when its first term can no longer be divided by the first term of the divisor.)