Factoring Polynomials

Factoring Numbers

• You will be learning in this lesson how to “undo” the process of multiplying polynomials that you learned to do from the previous lesson

• We will start by looking at factoring whole numbers

• We want to distinguish between two types of factoring: factorization of a number and prime factorization of a number

• These are related but not exactly the same

• Also remember that we are working here only with integers, so the numbers we will use and produce have no fraction or decimals parts

Factoring Numbers

• Factorization of a whole number means to find all factors of the number; that is, all numbers that divide into the original number with remainder zero

• For example, if asked to factor 12, you could simply list the numbers

• These are all, and the only whole numbers that divide evenly into 12 with no remainder

• What are the factors of 18?

Factoring Numbers

• By comparison, the prime factorization of a whole number means the set of primes that divide into the number with remainder zero

• This set of numbers is unique for each whole number

• Recall that prime numbers are those whose only factors are 1 and the number itself

• The primes less than 30 are

• In the prime factorization of a number, only these numbers are used (possibly with whole number exponents)

Factoring Numbers

• You saw that the factors of 12 are

• The prime factorization of 12 is

• So, while there are several ways to write 12 as a factor of two whole numbers

• there is only one way to write the prime factorization of 12 (not counting the order of the numbers):

Factoring Numbers

• Most students learn to prime factor a number by creating a factor tree

• You will not need this (most of the time) because you will be provided with a prime factorization table containing the prime factorization of whole numbers up to 194

• Save this table since we will use it for other lessons later in the semester!

Factoring a Monomial

• When factoring with expressions containing variables, we will consider each variable to be like a prime number

• That is, like a prime number, a variable will be considered divisible only by 1 and itself

• What this means is that a monomial such as is in prime factorization form

• We might imagine, however, that two other monomials were multiplied to create ; what might these have been?

Factoring a Monomial

• The monomials could be any combination (as products with exponents) of x and y such that the sum of the exponents of x add to 4 and the sum of the exponents of y add to 3

• For example,

• When factoring however, we do this in reverse

• If you asked to factor , you could write

• What are some other ways to factor ?

Guided Practice

Determine whether the factorization shown is correct. Use your factorization chart to factor the coefficients

a)

b)

c)

d)

e)

f)

Guided Practice

Determine whether the factorization shown is correct. Use your factorization chart to factor the coefficients

a) ; incorrect:

b) ; correct

c) ; correct

d) ; incorrect:

e) ; incorrect:

f) ; correct

Factoring a Monomial from a Polynomial

• To multiply a monomial times a polynomial, we use the distributive property once

• Factoring requires that we work this process in reverse

• A problem could ask, factor out the greatest common monomial from

• Our goal is to first find the greatest common monomial (red above), then to “undistributed” or factor it out

Factoring a Monomial from a Polynomial

Example: Completely factor the polynomial expression

• Use your prime factorization chart to rewrite coefficients:

• Note than in the first term there are 5 x multiplied, in the second term there are 4 x, and there are 3 x in the third term

• So every term has at least 3 x multiplied

• Also note that each term has at least

• The greatest common factor is

Factoring a Monomial from a Polynomial

• Now, factor each term such that is one of the factors

• Finally, use the distributive property in reverse to “factor out” the common factor

• We can express the coefficient as one number

Factoring a Monomial from a Polynomial

• Example: Completely factor the polynomial expression

• Note first of all that the last term is a constant; no variable is a factor

• This means that there is no variable common factor

• However, there is a coefficient common factor: use the prime factorization chart to factor each coefficient

Factoring a Monomial from a Polynomial

• Example: Completely factor the polynomial expression

• Next, note that each term has at least two factors 2; so (or 4) is the greatest common factor

• Finally, factor out the common factor

Factoring a Monomial from a Polynomial

• Example: Completely factor the polynomial expression

• Since the lead coefficient is and this cannot be factored, we have no coefficient common factor

• Note also that each term has at least 2 x’s multiplied; this is the greatest common factor

Factoring a Monomial from a Polynomial

• Let’s work this same problem again, but in a slightly different way

• It is sometimes desirable to factor out a negative common factor, as shown above

• Note that, in the last term, factoring out means you must also change the to

Factoring a Monomial from a Polynomial

• Now factor out

• We can always factor out a negative common factor, but we aren’t always required to do so

Guided Practice

Factor the greatest common factor (monomial) from each expression

a)

b)

c)

d)

e)

f)

Guided Practice

Factor the greatest common factor (monomial) from each expression

a)

b)

c)

d)

e) not factorable (prime)

f)

Practice 4

• Handout