Download topic: factoring polynomials

MAT029A
TOPIC: FACTORING POLYNOMIALS
Note some expressions don’t
factor and are called “prime.”
5.1 Introduction to Factoring
I. Preliminary Thoughts
A. Two concepts that get mixed up a lot are “LCM” and “GCF”
“Least Common Multiple”
vs
“Greatest Common Factor”
(numbers go into it)
(It goes into numbers)
Ex: LCM of 4,6 is 12
Ex: GCF of 4,6 is 2
2 3
3
Ex: LCM of x , x is x
Ex: GDF of x2, x3 is x2
Often found to add fractions
Often found to simplify fractions
GCF is used now!
B. To find the GCF with big numbers: Ex: What is the GCF of 40 and 100?
40
100
Step 1: Trees
2 20
2 50
2 10
2
25
2 5
5 5
Step 2: Multiply the primes they have in common.
II. This first sort of factoring is “Backward Distribution” so check by dist’n.
Ex: Factor out a monomial:
Step 1: Look at each coefficient to find the largest # that goes in and write it. It will be one
of the numbers or something smaller. (If it’s only the number one, skip it.)
Step 2: If any variable shows up in each term, find the smallest exponent and write it.
Step 3: Divide each term by this monomial and write the result in ( ).
2 - 2 - 4x
= 2x( - x – 2)
III. Factor “quadri-nomials” by “grouping”
Ex: Factor 8 + 2 + 12t + 3
Step 1: Drop parentheses around first two terms and then second two terms.
Careful of middle sign. Include it with the second grouping.
Step 2: Factor each set as above
Step 3: Write any common binomial in parentheses.
Step 4: Write “leftovers” in parentheses.
8 + 2 + 12t + 3
= (8 + 2 )( + 12t + 3)
= 2 (4t + 1) + 3(4t + 1)
=(4t + 1)(2 + 3)
5.2 Factoring Trinomials of the Type x2 + bx + c
I. If the coefficient of x2 is an invisible 1, consider “trial and error”. If you really hate it, 5.4
provides the most general strategy for factoring trinomials so go there now.
Ex: Factor + 3x + 2
Step 1: Write (x )(x )
Step 2: If sign of constant term is negative, then put different signs next.
If sign of constant term is positive,
then put two +’s if middle term is positive OR two –‘s if middle is negative.
Step 3: Look at the constant term.
If you wrote different signs, look for two factors of the last number with a
difference of the middle coefficient.
If you wrote same signs, look for two factors of the last number with a sum of the
middle coefficient.
+ 3x + 2
(x )(x )
(x + )(x + )
= (x + 2)(x + 1)
II. General factoring tips for all factoring situations:
A. Before you try anything fancy, write in descending order and look for backwards
distribution, above. (factor out a -1 if it helps.)
B. After you factor, check by multiplying!
C. See two variables? Place combo platter in middle!
5.3 Factoring ax2 + bx + c, a is not equal to 1: trial and error
Skip this and go straight to 5.4
5.4 Factoring ax2 + bx + c, a is not equal to 1 (or not!): “The ac-Method”
I. If you want to learn just one technique beyond backwards distribution, this is it! “ac/grouping”
Ex: Factor
Step 1: Name coefficients (in descending order) a, b, c
Step 2: Multiply a∙c
Step 3: Find all pairs of factors for this.
Step 4: Select pair whose sum is b.
Step 5: Replace the middle term of the original trinomial using these as coefficients. (Split
the middle.) Careful now…you are not done yet!
Step 6: Drop ( ) ( ) to start grouping as described in 5.1
-x–2
a=1 b=-1 c=-2
(a)(c)=(1)(-2)=-2
1 -2
-1 2
=
=
=
5.5 Factoring Trinomial squares and differences of squares.
I. These are two shortcuts that depend on pattern recognition.
A. Pattern recognition in binomial: “Diff. of Squares”
Ex: Factor
Step 1: Pattern Recognition:
Are both terms perfect squares? (Yes)
Subtraction? (Yes)
Step 2: Write: (
+
(
–4
= (x + 2)(x – 2)
Would split twice! Check it out!
B. Pattern Recognition In Trinomial : “Trinomial Square”
Ex: Factor
+ 6x + 9
Step 1: Pattern Recognition:
Are first and last terms perfect squares? (Yes)
Is the third term positive? (Yes)
Is sign-less middle term = 2
? (yes)
2
Step 2: Write: (
if middle is negative.
2
(
+
if middle is positive.
+ 6x + 9
=
5.6
Factoring: A general strategy
I. Generally it’s best to have descending order and a -1 factored out if the first term is negative:
Ex:
+ 2 = -1 (x2 - 3x - 2)
II. First Factoring to try is always “Backwards Distribution.” (5.1)
Ex:
III. Count number of terms.
A. Two terms? Try “Difference of Squares” shortcut (5.5)
Ex:
B. Four terms? Try “Grouping” (5.1)
Ex:
C. Three terms? {Note: If there are two variables, put the combo in the middle.}
1. Try “Trinomial Square” shortcut (5.5)
Ex:
2. Try “Trial and Error” shortcut (5.2)
Ex:
3. If these don’t work, try “ac/grouping” (5.4)
IV. See whether you can factor again (and again…)
V. Check factoring by multiplying.
5.7
Solving Quadratic Equations by Factoring
I. Earlier you learned how to solve “linear equations”; these equations can be recognized by the
fact that the unknown is not squared at any point and you usually get one solution. In
“quadratic equations” the unknown is squared at some point and you often get two solutions.
II. Steps to solve these are listed below. Easier problems don’t call for all steps. You may end up
jumping in later in the process!
Ex: solve x (x – 5) = 14
A. The form you want is Factored Stuff = 0. If not…
1. Send all terms to one side so you have “= 0” on one side.
2. Factor the other side. In order to do this, you may have to first multiply and/or write
terms in descending order.
B. Principal of Zero: If stuff that’s multiplied equals zero, then at least one factor is equal to
zero.
This means you should set each factor equal to zero.
Check…
III. On any graph, x-intercepts happen when y=0 so substitute 0 for y in a two variable quadratic
equation to find x-intercept on parabola.
5.8 Applications of Quadratic Equations
As in any word problem, Please Let Frankie Sing! (Picture/Label/Formulas/Substitutions)
I. Area of a rectangle (a = lw) and area of a triangle (a = ½ bh) should be familiar! Look inside
the cover of most texts for more formulas.
II. When you are given a formula, make sure you understand what variables represent and what
units are to be used.
III. Consecutive integers such as 2,3,4 can be represented using these expressions: x, x+1, x+2, etc.
Consecutive even or odd integers need these expressions: x, x + 2, x + 4, etc.
IV. Pythagorean Theorem: a2+b2=c2 if and only if you have a right triangle
(Make sure c represents the hypotenuse!)