Factoring Worksheet With Examples Page 17
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50Factoring Trinomials:
8-3
2
x
+ bx + c
Main Ideas
Tamika has enough bricks to make a
•
Factor trinomials of
2
the form x
+ bx + c.
30-foot border around the rectangular
vegetable garden she is planting. The
•
Solve equations of
2
the form
nursery says that the plants will need a
A
54 ft
2
x
+ bx + c = 0.
space of 54 square feet to grow. What
should the dimensions of her garden be?
P
30 ft
To solve this problem, you need to find
two numbers with a product of 54 and a
sum of 15, half the perimeter of the garden.
2
Factor x
+ bx + c
When two numbers are multiplied, each number is
a factor of the product. Similarly, when two binomials are multiplied,
each binomial is a factor of the product. To factor certain types of
trinomials, you will use the pattern for multiplying two binomials.
Study the following example.
F
O
I
L
(x + 2)(x + 3) = (x · x) + (x · 3) + (x · 2) + (2 · 3)
Use the FOIL method.
2
= x
+ 3x + 2x + 6
Simplify.
2
= x
+ (3 + 2)x + 6
Distributive Property
2
= x
+ 5x + 6
Simplify.
Observe the following pattern in this multiplication.
2
(x + 2)(x + 3) = x
+ (3 + 2)x + (2 · 3)
2
(x + m)(x + n) = x
+
(n
+ m)x +
mn
Let 2 = m and 3 = n.
2
= x
+
(m
+ n)x +
mn
Commutative (+)
2
x
+
bx
+ c
b = m + n and c = mn
Notice that the coefficient of the middle term is the sum of m and n
and the last term is the product of m and n. This pattern can be used
2
+ bx + c.
to factor trinomials of the form x
2
Factoring x
+ bx + c
2
Words
To factor trinomials of the form x
+ bx + c, find two integers,
m and n, with a sum equal to b and a product equal to c. Then
2
write x
+ bx + c as (x + m)(x + n).
2
Symbols
x
+ bx + c = (x + m)(x + n) when m + n = b and mn = c.
2
Example
x
+ 5x + 6 = (x + 2)(x + 3), since 2 + 3 = 5 and 2 · 3 = 6
434
Chapter 8 Factoring
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