Download Factor Out the Greatest Common Factor Factor by Grouping

Math 95
7.4 "Review of Factoring"
Objectives:
*
Factor out the greatest common factor.
*
Factor by grouping.
*
Factor the di¤erence of two squares.
*
Factor trinomials.
*
Test for factorability.
*
Factor the sum and di¤erence of two cubes.
*
Use substitution to factor trinomials.
Factor Out the Greatest Common Factor
The Greatest Common Factor (GCF)
kThe greatest common factor of a list of integers is the largest common factor of those integers.k
Recall:
A polynomial that cannot be factored is called a prime polynomial or an irreducible polynomial.
Example 1: (Factoring out the greatest common factor)
Factor the greatest common factor from each polynomial.
a) 9x4 y 2 12x3 y 3
b) 4a4 b2 + 6a3 b2 + 2a2 b
Factor by Grouping
Factoring by Grouping:
Step 1: Group the terms of the polynomial so that each group has a common factor
Step 2:
Factor out the common factor from each group
Step 3:
Factor out the resulting common factor
Example 2: (Factoring by grouping)
Factor by grouping.
a) 3x3 y
4x2 y 2
6x2 y + 8xy 2
b) 3ax2 + 3bx2 + a + 5bx + 5ax + b
Page: 1
Notes by Bibiana Lopez
Beginning and Intermediate Algebra by Gustafson and Frisk
7.4
Factor the Di¤erence of Two Squares
Factoring a Di¤erence of Two Squares:
F2
L2 =
WARNING!!!
(F = First Quantity, L = Last Quantity)
Recall that
F 2 + L2 = prime polynomial
Example 3: (Factoring a di¤erence of two squares)
Factor.
a) 4x6 y 8
49z 16
b) 2x2 + 6x + 9
x2
Now, we will discuss techniques for factoring trinomials. These techniques are based on the fact that the product of two
binomials is often a trinomial.
Factor Perfect-Square Trinomials
Factoring Perfect-Square Trinomials:
a2 + 2ab + b2 =
a2
2ab + b2 =
Example 4: (Factoring perfect-square trinomials)
Factor the following trinomials.
a) x2 + 10x + 25
b) 4x4
Page: 2
12x2 y 2 + 9y 4
Notes by Bibiana Lopez
Beginning and Intermediate Algebra by Gustafson and Frisk
7.4
Factor Trinomials of the Form x2+bx+c
We will consider any trinomials of the form x2 + bx + c where the coe¢ cient of the square is 1 (leading coe¢ cient)
Factoring Trinomials Whose Leading Coe¢ cients is 1:
To factor a trinomial of the form x2 + bx + c, …nd two numbers whose product is c and whose sum is b
1.
If c is positive, the numbers have the same sign.
2: If c is negative, the numbers have di¤erent signs.
Then write the trinomial as a product of two binomials.
Example 5: (Factoring trinomials of the form x2 + bx + c)
Factor the following trinomials.
a) 2x2 y 2 + 4xy 3
30y 4
b) t4
8t2 + 12
Factor Trinomials of the Form ax2+bx+c
Factoring Trinomials by Grouping (Leading Coe¢ cients Other Than 1)
To factor a trinomial by grouping
1.
2.
Factor out any GCF.
Identify a; b; and c, and …nd the key number ac:
3.
Find two integers whose product is the key number and whose sum is b:
4.
Express the middle term, bx; as the sum (or di¤erence) of two terms.
5.
Enter the two numbers found in step 2 as coe¢ cients of x in the form shown below.
6.
Factor the equivalent four-term polynomial by grouping.
Example 6: (Factoring trinomials of the form ax2 + bx + c)
Factor the following trinomials.
a) 5x2 + 7x + 2
b) 18a
Page: 3
6ap2 + 3ap
Notes by Bibiana Lopez
Beginning and Intermediate Algebra by Gustafson and Frisk
7.4
Test for Factorability
Test for Factorability:
A trinomial of the form ax2 + bx + c; with integer coe¢ cients and a 6= 0, will factor into two binomials
with integer coe¢ cients if the value of b2 4ac is a perfect square.
If b2 4ac = 0 the factors will be the same.
;
If b2
4ac is not a perfect square, the trinomial is prime.
Example 7: (Factoring trinomials of the form ax2 + bx + c)
Factor the following trinomials, if possible.
a) 4w2 + 20w + 3
b) b2n
bn
6
Factoring the Sum and Di¤erence of Two Cubes:
F 3 + L3 =
F3
L3 =
(F = First Quantity, L = Last Quantity)
Example 8: (Factoring the sum and di¤erence of two cubes)
Factor the following algebraic expressions.
a) x3 + 8
b) 8c6
125d3
Use Substitution to Factor Trinomials
For more complicated expressions, especially those involving a quantity within parentheses, a substitution sometimes
helps to simplify the factoring process.
Example 9: (Using substitution to factor trinomials)
Factor the following polynomial by using substitution.
2
(x + y) + 7 (x + y) + 12
Page: 4
Notes by Bibiana Lopez
Beginning and Intermediate Algebra by Gustafson and Frisk
7.4
Steps for Factoring a Polynomial:
Step 1: Is there a common factor?
If so, factor out the GCF, or the opposite of the GCF (so that the leading coe¢ cient is positive).
Step 2:
How many terms does the polynomial have?
If it has two terms, look for the following problem types:
a.
The di¤erence of two squares
b.
The sum of two cubes
c.
The di¤erence of two cubes
If it has three terms, look for the following problem types:
a.
A perfect-square trinomial
b.
Use grouping method
If it has four terms, try to factor by grouping
Step 3:
Can any factors be factored further?
Step 4:
Does the factorization check?
If so, factor them completely
Check by multiplying
Example 10: (Using the steps for factoring a polynomial)
Factor completely:
a) 2x2
11x + 5
b) a12
Page: 5
64
Notes by Bibiana Lopez