Transcript Finding Slope – Intercept Form From Points

Factoring Trinomials
Module VII, Lesson 5
Online Algebra
VHS@PWCS
Factoring
 Factors are integers that divide another integer
evenly.
 In a multiplication problem factors are the
numbers that are being multiplied to get the
product (the answer).
 What are the factors of 90?
1
x 90
 2 x 45
 3 x 30
 5 x 18
 9 x 10
If we multiply these factors
together the product is 90!
Factors
Multiply the following
binomials.
1. (x + 5)(x + 7)
x2 + 12x + 35
2. (2x – 3)(x + 4)
2x2 +5x -12
3. (x – 2)(3x – 1)
3x2 -7x + 1
(x + 5) and (x + 7) are
factors of x2 + 12x + 7.
(2x – 3) and (x + 4) are
factors of 2x2 + 5x – 12.
(x – 2) and (3x – 1) are
factors of 3x2 – 7x + 1
When we multiply the binomials above
we get a quadratic trinomial.
Factoring Trinomials
Factoring trinomials of the form x2 + bx + c,.
Remember that if there is no coefficient in front of
the x it is 1.
Examples of this type of trinomial are:
x2 + 4x - 3
x2 – 7x + 9
x2 + 6x + 26
Factoring Trinomials
To factor x2 + bx + c:
1. Find the factors of c
2. Find the sum of each pair of factors.
3. Use the factors of c that add up to b to
put into binomials as follows:
(x + one of the factors)(x + the other factor)
This looks kind of confusing so lets try it with numbers.
Factoring Trinomials
Factor x2 + 5x + 6
1.
2.
3.
Find the factors of c
Find the sum of each pair of
factors.
Use the factors of c that add
up to b to put into binomials
as follows:
(x + one of the factors)(x + the other factor)
 Factors of 6


6 and 1
2 and 3
 Sum of the factors


6+1=7
2+3=5
 (x + 2)(x + 3)
You can use FOIL to check:
(x + 2)(x + 3) = x2 + 3x + 2x + 6 = x2 + 5x + 6
Factoring trinomials
1.
2.
3.
4.
y2 – 4y – 45
Find the factors of -45. In this case since the c is
negative we need one negative factor and one positive
factor.
-1
45 45
= 44
1 1+and
-45 -45
= -44
-1+and
-3
15 15
= 12
-3+and
3 3+and
-15 -15
= -12
-5
+
9
=
4
-5 and 9
5 and -9
5 + -9 = -4
Find the sum of the factors.
Use the factors that add up to -4.
(y – 9)(y + 5)
You can use foil to check.
(y – 9)(y + 5) = y2 + 5y – 9y – 45 = y2 – 4y - 45
Try these on your own!
1. c2 – 2c + 1
1. (c – 1)(c – 1)
2. r2 + 6r – 16
2. (r – 2)(r + 8)
3. x2 + 10x + 25
3. (x + 5)(x + 5)
Do you notice any patterns?
 If b and c are positive, then the factors you will use are
both positive.
x2 + 10x + 25 = (x + 5)(x + 5)
 If b is negative and c is positive, then the factors you will
use are both negative.
x2 – 5x + 6 = (x -2)(x – 3)
 If b is negative and c is negative, then one factor will be
positive and the other will be negative. The negative
number must have a larger absolute value.
x2 - 5x – 6 = (x – 6)(x + 1)
 If b is negative and c is positive, then one factor will be
positive and the other will be negative. The positive
number must have a larger absolute value.
x2 + 5x – 6 = (x + 6)(x – 1)
Factoring Trinomials – Factor
by Grouping
To factor trinomials of the form ax2 + bx + c, we use what we call factor
by grouping.
1.
Find the product (multiply) of a and c.
2.
Find the factors of the product of a and c.
3.
Use the factors that add up to b.
4.
Write the quadratic as:
ax2 + (one of the factors)x + (the other factor)x + c.
5.
6.
Factor the first 2 terms, then the second 2 terms. The goal when
factoring is to get the same binomial.
Write as:
(same binomial)(GCF of First pair + GCF of second pair)
All this is pretty difficult to explain so we will do quite a few examples.
Factor: 5x2 – 2x - 7
5 x -7 = -35
Since the product is negative we
need one positive and one
negative, with the larger negative.
1 and – 35
5 and – 7
3. 5 + -7 = -2
4. 5x2 + 5x + -7x – 7
Notice that the middle two terms add
up to -2x, the middle term in the
trinomial.
5x + -7x = -2x
5. 5x(x + 1) – 7(x + 1)
5x is the GCF of 5x2 + 5x
-7 is the GCF of -7x – 7
x + 1 is the binomial left for both when
you pull out the GCF
6. (x + 1)(5x – 7)
1.
2.
1.
2.
3.
4.
Find the product (multiply) of a
and c.
Find the factors of the product of
a and c The patterns that we
found still apply.
Use the factors that add up to b.
Write the quadratic as:
ax2 + (one of the factors)x + (the other factor)x + c.
5.
6.
Factor the first 2 terms, then the
second 2 terms. The goal when
factoring is to get the same
binomial.
Write as:
(same binomial)(GCF of First pair + GCF of second pair)
Factor 3x2 + 13x - 10
1.
2.
3.
4.
Find the product (multiply) of a and
c.
Find the factors of the product of a
and c The patterns that we found
still apply.
Use the factors that add up to b.
Write the quadratic as:
ax2 + (one of the factors)x + (the other factor)x + c.
5.
6.
Factor the first 2 terms, then the
second 2 terms. The goal when
factoring is to get the same
binomial.
Write as:
(same binomial)(GCF of First pair + GCF of second pair)
1. 3 x -10 = -30
2. -1 and 30
3.
4.
5.
6.
-2 and 15
-3 and 10
-5 and 6
-2 + 15 = 13
3x2 + -2x + 15x – 10
x(3x – 2) + 5(3x – 2)
(x + 5)(3x – 2)
Factor: 3x2 + 17x + 20
1.
2.
3.
4.
Find the product (multiply) of a and
c.
Find the factors of the product of a
and c The patterns that we found
still apply.
Use the factors that add up to b.
Write the quadratic as:
1. 3 x 20 = 60
2. 1 and 60
ax2 + (one of the factors)x + (the other factor)x + c.
5.
6.
Factor the first 2 terms, then the
second 2 terms. The goal when
factoring is to get the same
binomial.
Write as:
(same binomial)(GCF of First pair + GCF of second
pair)
3.
4.
5.
6.
2 and 30
3 and 20
4 and 15
5 and 12
6 and 10
5 + 12 = 17
3x2 + 12x + 5x + 20
3x(x + 4) + 5(x + 4)
(3x + 5)(x + 4)
Try these on your own.
1. 2w2 – w – 3
2. 2t2 + 3t – 2
3. 6x2 + 10x + 4
1. 2w2 + 2w – 3w – 3
2w(w + 1) – 3(w + 1)
(2w – 3)(w + 1)
2. 2t2 – 1t + 4t – 2
t(2t – 1) + 2(2t – 1)
(t + 2)(2t – 1)
3. 6x2 + 6x + 4x + 4
6x(x + 1) + 4(x + 1)
(6x + 4)(x + 1)
Factoring Review
Remember that factors are:




Integers that divide another integer evenly
In a multiplication problem they are the
numbers that you multiply together
The factors of a quadratic trinomial are 2
binomials
Look for patterns!