Transcript ppt - Eudora Schools

Factoring
Trinomials
Section 6.2
MATH 116-460
Mr. Keltner
Monomials and
more monomials
 A monomial is an algebraic expression that is
either a number, a variable, or the product of a
number and a variable.
 A binomial, such as x - 7, is the sum of two
monomials.
 A trinomial, such as x2 + 8x +15, is the sum of
three monomials.
 A polynomial is the group that all of these
expressions belong to.

Monomials, binomials, and trinomials are just specific
types of polynomials.
Factoring Quadratic
Expressions
 When we worked with multiplying expressions
such as 2x and (x + 3), we wrote the product
as a sum, like 2x2 + 6x.

This used the Distributive Property.
 Factoring reverses the process, allowing us to
write the sum as a product.
 To factor an expression containing two or more
terms, try factoring out the greatest common
factor (GCF).
Example 1
 Factor each quadratic expression, by
finding the greatest common factor
(GCF) of the expression.

27c2 - 18c

5z(2z + 1) – 2(2z + 1)
Factoring algebraically
 Factor x2 + 3x – 10.
 Look for a pattern.

We will notice that the sums and products in the
x-term and constant term are related to the
factors of the last term in the factored expression
in the form (x -b)(x - c).
 This observation gives us a rule for factoring
quadratic expressions of the form x2 + bx + c.
Factoring
2
x
+ bx + c
 To factor an expression of the form
ax2 + bx + c
where a = 1, look for integers j and k such
that j•k = c and j + k = b.
 Then factor the expression.
x2 + bx + c
= x2 + (j + k)x + (j • k)
= (x + j) (x + k)
Example 2
 Factor each quadratic expression.

x2 + 12x + 27

n2 - 4n - 12

y2 + 10y - 24
If there’s another number
in front…
 Check for a monomial that can be
factored out of each term of the
expression.
 Example 3: Factor the expressions

3n3 - 18n2 + 24n

y5 + 3y4 - 18y3
Don’t get overwhelmed by
seeing several variables
 By keeping the FOIL method in mind and knowing
where the last term comes from, factoring trinomials
with more than one variable is easier than it looks.
 It is always a good idea to look for a monomial factor.
 Example 4: Factor the
expressions
 x2 + 9xy + 20y2

x3 + 4x2y - 21xy2
2
ax
Factoring
+ bx + c
when a≠1: Trial and Error
 In the case where our leading
coefficient is NOT 1 and we cannot
factor a monomial out of the
expression,consider


The factors of the ax2 term
The factors of the c term
 Keep in mind some behaviors of odd
and even numbers:
 Even + Even = Even = Odd + Odd 
 Even • Even = Even = Even • Odd 
 Odd • Odd = Odd 
Example 5: Factoring by
Trial and Error
 Factor each expression and check by
using the FOIL method.

6t2 - 19t + 10

56n4 - 70n3 + 21n2

6x3 - 28x2y - 48xy2
Factoring by
Grouping
 To factor a trinomial of the form ax2 + bx +
c, where a≠1, by using grouping:




Look for a monomial GCF in all the terms.
Factor it out, is possible.
Find two factors of ac (are divisible by one of
the numbers a or c) which happen to add up
to b.
Write the polynomial with four terms so that
the bx term is written as the sum of two like
terms with coefficients that we used in step 2.
Factor by grouping!
Factoring by
Grouping: Your turn
 Factor each expression by grouping.

10x2 - 19x + 6

36x4y + 3x3y - 60x2y
Using substitution for
something better to look at
Substitution
 Some rather ugly polynomials are actually in
the form ax2 + bx + c and can be factored by
using a method called substitution.

This takes advantage of taking a messy part of the
expression and replacing it with a single variable.
Example 7: Factoring by
Substitution
 Factor each expression by using
substitution:

24(t + 2)2 - 22(t + 2) + 3

15n8 - 19n4 + 6
Assessment
Pgs. 403-404:
#’s 10-85, multiples of 5