Download Algebra - mathemons

Topic: Special Products:
Square of a Binomial
Essential Question
How can special
products and factors help
determine patterns from
various real-life
situations?
Introduction
Man cannot live without a smoother
relationship with others. So that when two
persons are related to each other, their
relationship can be described in two
opposite ways. If Dr. Rubio is John’s
teacher, then we can also say that John is
Dr. Rubio’s student. This is the same true in
Algebra, numbers/expressions too are
related to each other. We can also say that
4 is related to 2 in manner that 4 is the
square of 2 and 2 is the square root of 4.
Special Products
In mathematics products are obtained
by multiplication. In this section, you
will discover patterns that help you
determine the products of polynomials.
These are called special products.
They are called special products
because products are obtained through
definite patterns.
Recall: Laws of Exponents
1. The Product of Powers
am ∙ an = am+n
Examples:
∙ =
4
5
9
x ∙x =x
3
x
2
x
5
x
Another Example
3
(2x )
4
(-3x )
7
-6x
=
2. The Power of a Power
(am)n = amn
Examples:
(x4)3 = x12
(x2)3 = x6
(4x3)2 = 16x6
Another Example
4
5
(3y z)
=
20
5
243y z
3. The Power of a Product
m
(ab)
=
m
m
a b
Examples:
(2x)3 = 8x3
(2a2b4c7)4 = 16a8b16c28
Another Example
4
5
2
(-5x y z)
8
10
2
25x y z
=
Square of a Binomial
2
(x+y)
2
(x-y)
Multiply. We can find a shortcut.
(x + y)2
This is the square of a binomial pattern.
(x + y) (x + y)
x² + xy
=
+
x² + 2xy + y2
This is a “Perfect Square Trinomial.”
xy
+
y2
Shortcut: Square the first term, add twice
the product of both terms and add the
square of the second term.
Multiply. Use the shortcut.
(4x + 5)2
Shortcut:
=
x² + 2xy + y2
(4x)² + 2(4x 5) + (5)2
●
=
16x² + 40x + 25
Try these!
(x + 3)2
x² + 6x + 9
(5m + 8)2
25m² + 80m + 64
(2x + 4y)2
4x² + 16xy + 16y²
(-4x + 7)2
16x²- 56x + 49
Multiply. We can find a shortcut.
(x – y)2
This is the square of a binomial pattern.
(x – y) (x – y)
x² - xy
=
-
x² - 2xy + y2
This is a “Perfect Square Trinomial.”
xy
+
y2
Multiply. Use the shortcut.
(3x - 7)2
Shortcut:
x² - 2xy + y2
=
9x² - 42x + 49
Try these!
(x – 7)2
x² - 14x + 49
(3p - 4)2
9p² - 24p + 16
(4x - 6y)2
16x² - 48xy + 36y²
Homework # 2