7. Multiplying monomials and polynomials

The lesson, step by step

1. Warm-up

   Ask: What is a³ written out? What is a² × a³ written out? Count the a's. (aaaaa, so a⁵.)

2. The rule of the exponents

   Read Arts. 67 to 69.

   The exponent of a letter in the product is equal to the sum of its exponents in the two factors.

   Then the rule for monomials:

   1. Multiply the coefficients of the two terms together.

   2. To this product, annex all the letters in both quantities.

   3. When the same letter occurs in both factors, add its exponents for the exponent of the product.

3. Guided practice: monomials

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

   7. Multiply 2bc by mn. Ans. 2bcmn.

   8. Multiply 4ab by 5xy. Ans. 20abxy.

   Then 3a²b by 4ab: 12a³b². Remind: write the letters in the order of the alphabet.

4. Polynomial times monomial

   Read the oranges and lemons example in Art. 70 (5 oranges and 2 lemons at 4 cents: 20 + 8 = 28 cents) and the rule:

   Multiply each term of the multiplicand by the multiplier.

   Work exercise 3 together: 4x + 5y times 3a is 12ax + 15ay.

5. Polynomial times polynomial

   Read Art. 71.

   Multiply each term of the multiplicand by each term of the multiplier, and add the products together.

   Work exercise 10 together, set up like long multiplication:

   10. 3x+2y by 2x+3y. . . Ans. 6x²+13xy+6y².

   The two middle products, 9xy and 4xy, sit under each other and add to 13xy.

6. Independent practice

   Art. 69 exercises 3 to 13, Art. 70 exercises 2 to 8, and Art. 71 exercises 4 to 15. Check each against the printed answer.

7. Check

   For one product, put x = 2 and y = 1 into the factors and into the answer; they must agree. Afterwards your student can confirm one answer on the site's calculator with expand((3*x+2*y)*(2*x+3*y)).

8. Wrap-up

   Your student explains why a² times a³ is a⁵ and not a⁶.