CHAPTER 5 – FACTORING
5.1 Common Factors
The ______of two or more monomials is the product of:
- The GCF of the numerical coefficients
- The common variable factor – which is the smallest exponent of each variable shared by the monomials
The GCF of is since the GCF of 30 and 18 is 6 and the smallest exponent on x is 2, the smallest exponent on y is 3.
Finding GCF’s of coefficients is a skill that improves with practice. If you are having trouble seeing the GCF, the general rule is to break the coefficients into a product of simpler factors, and take only those factors which appear on both factor lists.
Example: Find the GCF of 180 and 108
The two terms have common factors of 2, 2, 3, 3, so the GCF
Two terms which share no factors, such as 14 and 25, have a GCF of 1 and are said to be mutually prime.
To ______a polynomial expression means to rewrite the polynomial as a product of other polynomials which share no monomial factors.
Example: Factor
Alwaysbegin factoring by searching for a greatest common factor. If no GCF exists, we will have to use another of the methods examined later in the chapter.
Here the GCF = 3x, since the GCF of 24, 15, and 90 is 3, and the smallest exponent of x is 1. This GCF becomes one of two factors.
The second factor is found by dividing the GCF into each of the terms of the original polynomial.
So can be factored as . To check this, you can multiply it back out to see if the result is
Examples of factoring by method of GCF:
Special Note: If a factor is given of the form (b – a), it can be changed into (a – b) by removing the common factor of -1. Recall that dividing any expression by -1 will result in the opposite of the original expression.
Examples: (9 – x) = (-1) (x – 9)(5 – 2x) = (-1) (2x – 5)
Factoring Polynomials of four terms: Grouping
To factor a polynomial containing 4 terms where there is no GCF present, we can try to use a technique called grouping. It involves breaking the polynomial into 2 binomials (2 “groups”) and then looking for a GCF withing each new group.
Example: Factor
To begin, we should look for a greatest common factor. There is none.
Next, break the expression into two groups:
Now, check each group for a GCF.
The GCF of the first binomial is: ______Factor it out to get ______
The GCF of the second binomial is: ______Factor it out to get ______
Now, stringing the two groups back together, we get:
Note that the two groups each have the same term,, in their factorization.
This makesa common factor! Factor it out! What is left behind? ______
The result is a pair of binomial factors: FOIL it out to check:
Something to watch for when using grouping:
If the second operator you see is subtraction, when you create the second group, you must change the operator: addition subtraction, subtraction addition:
Examples:
(Finish it)
The trick is always, make the two binomial factors look exactly the same before finishing it off:
Examples:
In each case, we can FOIL to verify we have the correct factorization.
5.2 Factoring Polynomials of the Form
When we talk about trinomials of the form , b and c refer to constants:
Example: has b = -5 and c = 4
To factor a trinomial of this form, we need to recognize how the trinomial could be found by multiplication of two factors. Since it shares no common factors, it could only be a result of multiplying two ______together by ______.
Example: =
The trick is, how do we find the two binomial factors?
Look at the result above.
Note that the b value, -5, came from doing what to -1 and -4? ______
What about the c value, +4, as it relates to -1 and -4? ______
IDEA: To factor a trinomial of the form we are looking for two binomial factors, call them such that:
- b = the sum of m and n
- c = the product of m and n
Important Relationships:
1. Since c = m n if c is positive, m and n must have the same sign, and
if c is negative, m and n must have opposite signs
2. The sign of b will determine the signs of m & n if c is positive
3. When c is negative, the sign of b will also be the sign of the larger number (m or n)
Example: Since b was +, the 10 gets +
Since b was – , the 10 gets –
Method of finding the correct pair of factors, (x+m) (x+n)
- Find all possible pairs of factors of c, using the guidelines on the previous page to determine what the signs should be.
- Find the pair of factors which ADDS up to b.
Example: Factor
Since the c is +, m & n must have the same sign. The sign is –, since b is negative.
So we are looking for two negative factors of 15 which add up to –8.
Factors of –15Sum
-1, -15-16
-3, -5-8 Here are two factors of –15 which add to –8
factors as
Example: Factor
Since c is –, we know that m & n have opposite signs, with the larger of the two factors getting a – sign, because b is also negative.
We are looking for two factors of –66 which add up to –5.
Factors of –66Sum
+1, -66-65
+2, -33-31
+3, -22-19
+6, -11-5 Here are two factors of –66 which add to –5 !!
factors as Note: Order of the factors is not important.
You try one!!
Example: Factor
Factors of –30Sum
Examples of factoring trinomials of the form
5.3 Factoring trinomials of the form
When the lead coefficient on the x-squared term is not one, we need to use a different approach.
Here we cannot claim that b is formed completely of the two factors of c. The factors of a will also contribute to the b value.
Example:
As you can see, the b value of –29 is the sum of two numbers, each of which is a product of two factors: -14 is the product of the 2 and the –7, -15 is the product of the 3 and the –5. If we take –29 and break it into two pieces, -14 and –15, watch what happens:
It becomes a GROUPING PROBLEM!!
The trick here is this: We have to be able to break up the –29 into the correct two pieces to get the grouping procedure to work! What we would like to do is find the correct two terms to break b into and perform a reverse FOIL.
Here’s how we do it: Start by recognizing that b is formed by a special combination of the four factors which make up a and c. So we will get one big group of factors by multiplying a and c.
- Get ac (the product of the constant and the coefficient on the x-squared term)
- Go about finding two factors of ac which add up to b, as we did before.
- These two number which add to b will be used to replace the single bx term with two terms, transforming our TRINOMIAL into a 4-TERM polynomial
- Now factor the result by GROUPING.
Example: Factor
- Get the product ac = 2(3) = 6
- Find a pair of factors of 6 which add up to –7. The two factors can be found using the table approach: Factors of 6 Sum
-1, -6-7 Here are the two correct factors
-2, -3-5
- The two factors, -1 and -6, are used to replace the original value of –7 in the polynomial. becomes
- Use grouping to factor the 4-term polynomial.
Check the result by FOILing it out.
Examples of factoring trinomials of the form
5.4 Special Factors
Factoring the difference of two perfect squares:
In the previous chapter, we saw that when we multiply the sum and difference of two monomials, the result is a difference of their squares.
Logically, we should also be able to work in reverse in a factoring problem.
The difference of two perfect squares should factor into the sum and difference of the square roots of the two monomial terms of the binomial.
Example: factors into
Examples:
Factoring Completely
When asked to factor completely, what is being hinted at is that there will be a need for more than one step in the factoring process.
- Always start by trying to find a GCF!!
- The remaining factorization depends on the form:
- Four terms – grouping
- Trinomial – use the corresponding or method
- Binomial – check for difference of perfect squares
Then difference of perfect squares
Example: Factor: First use GCF:
Examples:
5.5 Solving Quadratic Equations by Factoring
Principle of Zero Products:
If the product of two factors is zero, then at least one of the factors must equal zero.
If then either , , or both
Example:
If and we would like to solve for x, we need to apply the principle.
Since the product can only be zero if one or both of the factors is zero, we can make the equation true by finding the values that make the two factors equal to zero.
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Therefore the equation has two solutions: 7 & -9
Our goal in solving equations is to rewrite them in factored form where possible, and then set each of the individual factors equal to zero. This will create two equations which are very easy to solve.
Quadratic Equations
Any equation of the form where a, b, and c are real numbers and a 0 is said to be a QUADRATIC EQUATION in standard form.
One way to solve a quadratic equation is to factor the quadratic trinomial and apply the principle of zero products. To use this method, though, it is important to note that the equation must be written in standard form.
Example: Solve
Example: Solve
Example: Solve
Examples: