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Algebra 2
Multiplying Polynomials
Warm-Up Problems
1. Simplify each expression.
a. (–4a3 + a2 –1) – (–3a3 – a2 + 2a + 5)
b. (3a2 – a –6) + (–a2 – a + 3)
2. Find –2b(3b3 – b2 + 5b – 6)
Vocabulary
• Binomial
– An expression consisting of two terms or monomials separated
by a plus (+) or minus (−) sign. Examples of binomials include ax
+ b, x2 − y2 and 2x + 3y. Even x + 2 + 7 is a binomial, since it
reduces to x + 9, which has two terms.
• Trinomial
– An algebraic expression consisting of three terms connected by
plus or minus signs.
• Polynomial
– An algebraic expression of two or more terms connected by plus
or minus signs.
Different Methods of Multiplying
Binomials
•
•
•
•
•
“Distributive” Method
“FOIL”
“Vertical” Method
“Grid” Method
Algebra Tiles
"Distributive" Method
The most universal method. Applies to all polynomial multiplications, not just to binomials.
Multiply (x + 3)(x + 2)
Step 1: Multiply the first term of the first
binomial with the first term of the second
binomial, remembering that when multiplying
the values to multiply the bases and add the
exponents.
Step 2: Multiply the first term of the first
binomial with the second term of the second
binomial.
Step 3: Multiply the second term of the first
binomial with the first term of the second
binomial.
Step 4: Multiply the second term of the first
binomial with the second term in the second
binomial.
Step 5: Put all four values into a single
equation by combining like terms.
Use the “Distributive” Method to
Solve:
The “FOIL” Method
Warning: This method can only be used with binomials!
The “Vertical” Method
(2x - 7)(x + 3)
21
×32
____
42
63
____
672
2x − 7
x+3
___________
6x − 21
2x2 − 7x
___________
2x2 − x − 21
(d - 1)(5d - 4)
The “Grid” Method
To multiply by the grid method, place one
binomial at the top of a 2x2 grid (for
binomials) and the second binomial on the
side of the grid. Place the terms such that
each term with its sign lines up with a row or
column of the grid. Multiply the rows and
columns of the grid to complete the interior of
the grid. Finish by adding together the entries
inside the grid.
(3x - 1)(2x + 3)
Answer: x2 + 5x + 6
Algebra Tiles
To multiply binomials using algebra tiles, place
one expression at the top of the grid and the
second expression on the side of the grid. You
MUST maintain straight lines when you are
filling in the center of the grid. The tiles
needed to complete the inner grid will be your
answer.
(x + 3)(x + 2)
Try It:
1. Use the “Grid” Method to solve (3x – 7)(x + 3).
2. Use Algebra Tiles to solve (2x + 4)(x + 3)
Squaring a Binomial
(x – 5)²
We know that (x – 5)² is (x – 5)(x – 5)
Try It:
Multiplying Polynomials
Try It:
Word Problem
The diagram shows an area rug on a hardwood floor with dimensions x and x + 3.
The rug leaves a border around the outside of the room. The wider borders are 2 feet
wide, the narrower ones on the sides are 1 foot wide.
a) What is an algebraic expression for the area of the floor? (hint: A = l × w)
b) What is an algebraic expression for the area of the rug?
c) What algebraic expression represents the area of the border?
d) If the value of x is 9 feet, find numerical values for parts a), b) and c).
Journal Entry
Please answer the following question in your
“warm-up” books:
Which way do you think is the easiest way to
multiply binomials: distributive method, FOIL,
vertical method, grid method, or algebra tiles?
You must explain your reasoning for your
decision.
Additional Practice Problems
Exercises
Try the following exercises:
1. (5x − 7)(x + 2)
2. (x2 + 3)(x2 + 3)
3. (x − y)(3x − 2y)
4. (4 − y)(y + 4)
5. (a + b)(c + d)
a) (x – 7) 2 =
b) (3a – 2b) 2 =
c) (2x + y) 2 =
d) (2x 3 + 4) 2 =
e) (3y – 5z) 2 =
f) (6a + 5b) 2 =
a) (2x + 3)(4x – 5) =
b) (2 – 7x)(9 + 2x) =
c) (9m – 2n)(3m – n) =
d) (5x – 3y)(2x + 9y) =
e) (xy + 1)(3xy – 1) =
f) (2x + 3n)(2x – 3n) =
g) (4c – 5d)(c + 2d) =
h) (3m2 – 2n2)(2m2 – n2) =
i) (6t + 1)(3t – 2) =
j) (2m + 3t)(3m – 4t) =