Transcript Chapter 4: Factoring Polynomials

Factoring
Polynomials
The Greatest Common
Factor
Factors
Factors
Factors are numbers (or polynomials) you can
multiply together to get another number (or
polynomial).
Factoring – writing a polynomial as a product of
polynomials.
Martin-Gay, Developmental Mathematics
3
Greatest Common Factor
Prime Numbers – numbers greater than one, have
only factors of 1 and itself
Composite Number – numbers greater than on e and
have more than 2 factors
Greatest common factor – largest quantity that is a
factor of all the integers or polynomials involved.
Finding the GCF of a List of Integers or Terms
1) Prime factor the numbers.
2) Identify common prime factors.
3) Take the product of all common prime factors.
• If there are no common prime factors, GCF is 1.
Martin-Gay, Developmental Mathematics
4
Greatest Common Factor
Example
Find the GCF of each list of numbers.
1) 12 and 8
12 = 2 · 2 · 3
8=2·2·2
So the GCF is 2 · 2 = 4.
2) 7 and 20
7=1·7
20 = 2 · 2 · 5
There are no common prime factors so the
GCF is 1.
Martin-Gay, Developmental Mathematics
5
Greatest Common Factor
GCF of monomials or terms –
1. write the gcf of the coefficients.
2. write the variables they have in common.
3. Select the lowest exponent for each variable
1)
4x3 and 6x7
4x3 = 2 · 2 · x · x · x
6x7 = 2 · 3 x · x · x · x · x · x · x
So the GCF is 2 · x · x · x = 2x3
Martin-Gay, Developmental Mathematics
6
Greatest Common Factor
GCF of monomials or terms –
1. write the gcf of the coefficients.
2. write the variables they have in common.
3. Select the lowest exponent for each variable
all the integers or polynomials involved.
1)
2)
8y3z + 12y2z
8y3z = 2 · 2 · 2 · y · y · y · z
12y2 z = 2 · 2 · 3 ·y · y · z
So the GCF is 2 · 2 · y · y · z = 4y2 z
Martin-Gay, Developmental Mathematics
7
Factoring Polynomials
The first step in factoring a polynomial is to
find the GCF of all its terms.
Then we write the polynomial as a product by
factoring out the GCF from all the terms.
Martin-Gay, Developmental Mathematics
8
Factoring out the GCF
Factor out the GCF in each of the following
polynomials.
1) 6x3 – 9x2 + 12x =
1
3x ( 2x2 –3x + 4 ) = 6x3 – 9x2 + 12x
GCF
what is the largest number that divides evenly into 6, 9 & 12?
Is there a variable that each term has in common?
what is the smallest exponent for the x of all 3 terms?
That is the GCF
Now open a set of parentheses and divide the polynomial by
the GCF to find the other factor
Check by multiplying the GCF and the other factor
Martin-Gay, Developmental Mathematics
9
Factoring out the GCF
Factor out the GCF in each of the following
polynomial.
2) 14x4 + 7x3 – 21x2 =
2
7x ( 2x2 + x – 3 ) = 14x4 + 7x3 – 21x2
GCF
what is the largest number that divides evenly into 14, 7 & 21?
Is there a variable that each term has in common?
what is the smallest exponent for the x of all 3 terms?
That is the GCF
Now open a set of parentheses and divide the polynomial by
the GCF to find the other factor
Check by multiplying the GCF and the other factor
Martin-Gay, Developmental Mathematics
10
Factoring
Remember that factoring out the GCF from the terms of
a polynomial should always be the first step in factoring
a polynomial.
This will usually be followed by additional steps in the
process.
Martin-Gay, Developmental Mathematics
11
Factoring Trinomials of the
2
Form x + bx + c
Factoring Trinomials
Recall by using the FOIL method that
F
O
I
L
(x + 2)(x + 4) = x2 + 4x + 2x + 8
= x2 + 6x + 8
To factor x2 + bx + c into (x + one #)(x + another #),
note that b is the sum of the two numbers and c is the
product of the two numbers.
So we’ll be looking for 2 numbers whose product is
c and whose sum is b.
Note: there are fewer choices for the product, so
that’s why we start there first.
Martin-Gay, Developmental Mathematics
13
Factoring Polynomials
Example
ax  bx  c
2
Factor the polynomial 1. x2 + 13x + 30.
30
3
10
Factor out GCF
1
2.
Multiply a c and put at top of he X
13 1
3.
Put b at the bottom of X
4.
Find the pair of factors of 30 that add to 13
5.
Put each of the factors over a
Positive
Factors
of 30 (DenominatorX
Sum of factors
6.
Write
the factors.
+ numerator)
1 30
31
(1x  3)(1x  10)
2 15
17
3 10
13
2
1.
( x  3)( x  10)  x  13x  30
Martin-Gay, Developmental Mathematics
14
Factoring Polynomials
Example
ax  bx  c
2
Factor the polynomial 2. x2 – 11x + 24.
24
-3
-8
Factor out GCF
1
2.
Multiply a c and put at top of he X
-11 1
3.
Put b at the bottom of X
4.
Find the pair of factors of 24 that add to -11
5.
Put each of the factors over a
6.
Write the factors. (DenominatorX + numerator)
Factors of 24
Sum of factors
(1x  3)(1x  8)
-1 -24
-25
-2 -12
-14
1.
-3
-8
2
(
x

3
)(
x

8
)

x
 11x  24
-11
Martin-Gay, Developmental Mathematics
15
Factoring Polynomials
Example
ax  bx  c
2
Factor the polynomial 3. x2 – 2x – 35.
-35
5
-7
Factor out GCF
1
2.
Multiply a c and put at top of he X
-2 1
3.
Put b at the bottom of X
4.
Find the pair of factors of -35 that add to -2
5.
Put each of the factors over a
Positive
Factors
of -35(DenominatorX
Sum of factors
6.
Write
the factors
+ numerator)
-1 35
34
(1x  5)(1x  7)
1 -35
-34
-5 7
2
2
5 -7
-2 ( x  5)( x  7)  x  2 x  35
1.
Martin-Gay, Developmental Mathematics
16
Prime Polynomials
Example
Factor the polynomial 4. x2 – 6x + 10.
Since our two numbers must have a product of 10 and a
sum of – 6, the two numbers will have to both be negative.
Negative factors of 10
Sum of Factors
– 1, – 10
– 11
– 2, – 5
–7
Since there is not a factor pair whose sum is – 6,
x2 – 6x +10 is not factorable and we call it a prime
polynomial.
Martin-Gay, Developmental Mathematics
17
Check Your Result!
You should always check your factoring
results by multiplying the factored polynomial
to verify that it is equal to the original
polynomial.
Many times you can detect computational
errors or errors in the signs of your numbers
by checking your results.
Martin-Gay, Developmental Mathematics
18
Factoring Trinomials of
2
the Form ax + bx + c
Factoring Trinomials
Returning to the FOIL method,
F
O
I L
(3x + 2)(x + 4) = 3x2 + 12x + 2x + 8
= 3x2 + 14x + 8
To factor ax2 + bx + c into (#1·x + #2)(#3·x + #4),
note that a is the product of the two first coefficients,
c is the product of the two last coefficients and b is
the sum of the products of the outside coefficients
and inside coefficients.
Note that b is the sum of 2 products, not just 2
numbers, as in the last section.
Martin-Gay, Developmental Mathematics
20
Factoring Polynomials
Example
ax  bx  c
2
Factor the polynomial 7. 3x2 + 5x + 2.
6
2
3 1
Factor out GCF
3
3 1
2.
Multiply a c and put at top of he X
5
3.
Put b at the bottom of X
4.
Find the pair of factors of 6 that add to 5
5.
Put each of the factors over a
6.
Write
(DenominatorX
+ numerator)
Factors
of the
6 factors
Sum
of factors
1
6
7
(3x  2)(1x  1)
2
3
5
-1 -6
-7
2
(
3
x

2
)(
x

1
)

3
x
 5x  2
-2 -3
-5
1.
Martin-Gay, Developmental Mathematics
21
Factoring Polynomials
Example
ax  bx  c
2
Factor the polynomial 8. 5d2 + 6d – 8.
-40
10 2
-4
Factor out GCF
5 1
5
2.
Multiply a c and put at top of he X
6
3.
Put b at the bottom of X
4.
Find the pair of factors of -40 that add to 6
5.
Put each of the factors over a
Factors
of the
- 40factors (DenominatorX
Sum of factors+ numerator)
6.
Write
1
-40
-39
2
-20
-18 (5d  4)( d  2)
4
-10
-6
2
(
5
d

4
)(
d

2
)

5
d
 6d  8
-4
10
6
1.
Martin-Gay, Developmental Mathematics
22