Download Lesson 3 – Multiplication of two Binomials

Lesson 3 – Multiplication of two Binomials
Specific Outcome:
– Model multiplication of two given binomials, concretely or pictorially, and record process symbolically (4.1).
– Relate multiplication of 2 binomial expressions to an area model (4.2).
– Explain, using examples, the relationship between multiplication of binomials and multiplication of two-digit numbers (4.3).
– Multiply two polynomials symbolically, and combine like terms (4.5).
– Identify and explain errors in a solution for a polynomial multiplication (4.7).
DEFINITIONS
Expand: Writing a product of polynomial factors as a polynomial.
e.g., 2𝑥(𝑥 − 3) = 2𝑥 2 − 6𝑥
Factors: To write as a product; for example 20 = 2 × 2 × 5
Product: The result when two or more numbers are multiplied; or the expression of one
number multiplied by another.
USING ALGEBRA TILES:
Consider: (x + 2)(x + 3)
How do we expand these 2 factors using algebra tiles to determine its product?
x+3
x+2
Example 1: Complete the algebra tiles diagrams and determine the binomials products.
a)
b)
c)
Practice Questions: Page 166 # 4(b, c), 5(b, d)
USING AN AREA MODEL:
The algebra tile diagram used to model (x + 2)(x + 3) on the previous page, can be
modified into an area model, which shows that the product of 2 binomials is equivalent
to 4 monomial products.
Algebra Tiles
x+3
Area Model
x+2
(x + 2)(x + 3) = x2 + 5x + 6
(x + 2)(x + 3)
= x2 + 3x + 2x + 6
= x2 + 5x + 6
Example 2: Use an area model to determine the product of each of the following
binomials.
a) (5𝑥 − 6)(2𝑥 + 1)
b) (𝑎2 − 5)(𝑎2 − 8)
c) (3𝑝 + 2𝑞)(𝑝 + 9𝑞)
d) (𝑎 + 𝑏)(𝑐 + 𝑑)
Note: An area model can be used to show that the multiplication of 2 two-digit numbers
can be performed as four separate products.
Consider:
32 x 34
Example 3: Use an area model to determine the following products. Verify with a
calculator.
a) 43 × 51
b) 76 × 82
Practice Questions: Page 166 # 9, 12(a, e, g) (Use area model only)
USING THE DISTRIBUTIVE PROPERTY:
Investigate: Use an area diagram to determine the product of the following binomials.
(𝑎 + 𝑏)(𝑐 + 𝑑)
Distributive Property for Binomials
(𝒂 + 𝒃)(𝒄 + 𝒅) = 𝒂(𝒄 + 𝒅) + 𝒃(𝒄 + 𝒅)
= 𝒂𝒄 + 𝒂𝒅 + 𝒃𝒄 + 𝒃𝒅
Example 4: Use distributive property to expand and simplify the following expressions.
a) (𝑥 + 3)(𝑥 + 2)
b) (𝑎 − 7)(2𝑎 − 1)
c) (𝑥 − 1)(2𝑥 + 3)
USING THE FOIL METHOD:
In the FOIL method, each letter stands for a product:
F – first term in each bracket multiplied
O – outside terms multiplied
I – inside terms multiplied
L – last term in each bracket multiplied
Example 5: Use FOIL to determine each product.
a) (𝑥 + 6)(𝑥 + 4)
b) (2𝑥 − 1)(5𝑥 + 3)
c) (𝑦 + 1)(3𝑦 − 2)
d) (6𝑥 − 𝑦)2
Example 6: The area of the rectangle shown can be written in the
form of ax2 + bx + c.
Write the value of a in first box.
Write the value of b in second box.
Write the value of c in third box.
x+3
2x + 1
Practice Questions: Page 177 # 8, 9 – 10 (a, c, e’s)