15. Greatest common divisor of polynomials

The lesson, step by step

1. Warm-up

   Factor x² + 2x − 3 and x² + 5x + 6. ((x + 3)(x − 1) and (x + 2)(x + 3).) Ask: What factor do they share?

2. The easy way: factor first

   Read note 7 of the rule:

   7. Since the G.C.D. of two or more quantities contains all the factors common to these quantities, it may often be found most easily by separating the quantities into factors.

   So the warm-up pair has G.C.D. x + 3. That is exercise 8.

3. The division method

   Read the rule:

   1. Divide the greater polynomial by the less, and if there is no remainder, the less quantity will be the divisor sought.

   2. If there is a remainder, divide the first divisor by it, and continue to divide the last divisor by the last remainder, until a divisor is obtained which leaves no remainder; this will be the greatest common divisor of the two given polynomials.

   SayIt is the same as the 116 and 36 example, but with polynomials.

4. Look at the worked examples

   Read examples 1 to 3 on pp. 59 and 60. Point out the two tricks (Arts. 104 and 105): a factor in only one quantity may be canceled, and the dividend may be multiplied by a number to avoid fractions.

5. Guided practice

   Your student does these while you watch, trying factoring first:

   8. x²+2x— 3 and x²+5x+6 Ans. x+3.

   Then exercise 5: 5a² + 5ax and a² − x². (5a(a + x) and (a + x)(a − x), so a + x.)

6. Independent practice

   Exercises 6, 9, 10 and 13. Use factoring where it works; use division for the others.

7. Check

   Divide each original quantity by the G.C.D. found. Both must come out exactly.

8. Wrap-up

   Ask: How do you find the G.C.D. of three quantities? (Note 6: find it for two, then for that result and the third.)