Aim #45: How do we factor perfect square trinomials and DOTS expressions?
HW: Handout
Do Now: Simplify each expression.
1) (x - 3)(x + 3)
2) (x + 4)(x + 4)
3) (2x + 5)(2x - 5)
4) (6 - 5y)
2
What is a perfect square? How can you identify perfect squares?
Are there polynomial expressions that are perfect squares? That is, are there
polynomials that were once the product of the same two terms?
Which expressions in the Do Now are perfect square trinomials? What must be
true of both the leading term and the constant term?
Which expressions in the Do Now are the product of a sum and difference of the
same two terms? What do you notice happens when the two binomials are
multiplied?
Find the general rule for squaring a binomial that results in a perfect square
trinomial.
2
2
a) (a + b)
b) (a - b)
c) Take a look at the perfect square trinomials that are your answers from parts
a and b. How does the middle term relate to the first and last term?
1) In each case, identify if the trinomial is a perfect square. If so, factor the
perfect square as a product of the same two terms.
2
2
a) x - 8x + 16
b) 4y + 20y + 25
2
d) 12 - 6g + g
e) 9 - 6x + x
4
25
c)
2
x +
4
15
x+
1
9
2
2
2) Find the value of c if the expression x + 14x + c is a perfect square trinomial.
2
3) Find the value of b if the expression 9x + bx + 25 is a perfect square trinomial.
2
4) Find the value of a if the expression ax + 12x + 9 is a perfect square trinomial.
Let us re-visit question 1 from the Do Now:
(x - 3)(x + 3)
2
x + 3x - 3x - 9
2
x + 0x - 9
2
x -9
or
x
-3
2
-3x
x x
3 3x
-9
When you multiply the sum and difference of the same two terms, the middle term
of the trinomial cancels out.
2
The binomial x - 9 is known as the:
Difference Of Two Squares (DOTS)
DOTS expressions are easily factorable into the product of two binomials which are
the sum and difference of the same two terms.
4) Factor the following binomials using the DOTS method.
2
2
a) x - 49
d) 9 - x
6
10
2
16
c) 25x - 16
14
2
e) y - x
6
g) 36n - 169m
j) x -
4
b) 4x - 121
1
100
h) .01x
100
f) -1 + y
50
8 4
k) r s - 144t
2
m) x - 0.64
8 36
p) -400 + c d
14 4
- .16y
2 4
2
l) 1 - 256n
2
6
n) 196x y - 9n m
4
4
q) x - 64y
6 2
i) w x - y z
o)
4
25
2
10
18
r -s
2
2
2
r) 25a - 49b m
5) Explain why the expressions are NOT factorable by the DOTS method.
2
7
a) x + 144
b) x - 49
4
c) ax - 100
Sum it up!
To identify if a trinomial is a perfect square trinomial:
• Both the leading term and the constant term must be perfect squares.
• The middle term must be double the product of a and b.
To identify if a binomial is factorable using the DOTS method:
• Both terms are the difference of two perfect squares.
DOTS expressions are NOT perfect squares. Regardless of what factoring
process needs to be done, we should always look to factor out the GCF first,
if possible.