factoring polynomials. factoring numbers you will be learning in this lesson how to “undo” the...
TRANSCRIPT
![Page 1: Factoring Polynomials. Factoring Numbers You will be learning in this lesson how to “undo” the process of multiplying polynomials that you learned to](https://reader036.vdocument.in/reader036/viewer/2022062518/56649ee45503460f94bf371c/html5/thumbnails/1.jpg)
Factoring Polynomials
![Page 2: Factoring Polynomials. Factoring Numbers You will be learning in this lesson how to “undo” the process of multiplying polynomials that you learned to](https://reader036.vdocument.in/reader036/viewer/2022062518/56649ee45503460f94bf371c/html5/thumbnails/2.jpg)
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
![Page 3: Factoring Polynomials. Factoring Numbers You will be learning in this lesson how to “undo” the process of multiplying polynomials that you learned to](https://reader036.vdocument.in/reader036/viewer/2022062518/56649ee45503460f94bf371c/html5/thumbnails/3.jpg)
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?
![Page 4: Factoring Polynomials. Factoring Numbers You will be learning in this lesson how to “undo” the process of multiplying polynomials that you learned to](https://reader036.vdocument.in/reader036/viewer/2022062518/56649ee45503460f94bf371c/html5/thumbnails/4.jpg)
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)
![Page 5: Factoring Polynomials. Factoring Numbers You will be learning in this lesson how to “undo” the process of multiplying polynomials that you learned to](https://reader036.vdocument.in/reader036/viewer/2022062518/56649ee45503460f94bf371c/html5/thumbnails/5.jpg)
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):
![Page 6: Factoring Polynomials. Factoring Numbers You will be learning in this lesson how to “undo” the process of multiplying polynomials that you learned to](https://reader036.vdocument.in/reader036/viewer/2022062518/56649ee45503460f94bf371c/html5/thumbnails/6.jpg)
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!
![Page 7: Factoring Polynomials. Factoring Numbers You will be learning in this lesson how to “undo” the process of multiplying polynomials that you learned to](https://reader036.vdocument.in/reader036/viewer/2022062518/56649ee45503460f94bf371c/html5/thumbnails/7.jpg)
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?
![Page 8: Factoring Polynomials. Factoring Numbers You will be learning in this lesson how to “undo” the process of multiplying polynomials that you learned to](https://reader036.vdocument.in/reader036/viewer/2022062518/56649ee45503460f94bf371c/html5/thumbnails/8.jpg)
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 ?
![Page 9: Factoring Polynomials. Factoring Numbers You will be learning in this lesson how to “undo” the process of multiplying polynomials that you learned to](https://reader036.vdocument.in/reader036/viewer/2022062518/56649ee45503460f94bf371c/html5/thumbnails/9.jpg)
Guided Practice
Determine whether the factorization shown is correct. Use your factorization chart to factor the coefficients
a)
b)
c)
d)
e)
f)
![Page 10: Factoring Polynomials. Factoring Numbers You will be learning in this lesson how to “undo” the process of multiplying polynomials that you learned to](https://reader036.vdocument.in/reader036/viewer/2022062518/56649ee45503460f94bf371c/html5/thumbnails/10.jpg)
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
![Page 11: Factoring Polynomials. Factoring Numbers You will be learning in this lesson how to “undo” the process of multiplying polynomials that you learned to](https://reader036.vdocument.in/reader036/viewer/2022062518/56649ee45503460f94bf371c/html5/thumbnails/11.jpg)
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
![Page 12: Factoring Polynomials. Factoring Numbers You will be learning in this lesson how to “undo” the process of multiplying polynomials that you learned to](https://reader036.vdocument.in/reader036/viewer/2022062518/56649ee45503460f94bf371c/html5/thumbnails/12.jpg)
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
![Page 13: Factoring Polynomials. Factoring Numbers You will be learning in this lesson how to “undo” the process of multiplying polynomials that you learned to](https://reader036.vdocument.in/reader036/viewer/2022062518/56649ee45503460f94bf371c/html5/thumbnails/13.jpg)
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
![Page 14: Factoring Polynomials. Factoring Numbers You will be learning in this lesson how to “undo” the process of multiplying polynomials that you learned to](https://reader036.vdocument.in/reader036/viewer/2022062518/56649ee45503460f94bf371c/html5/thumbnails/14.jpg)
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
![Page 15: Factoring Polynomials. Factoring Numbers You will be learning in this lesson how to “undo” the process of multiplying polynomials that you learned to](https://reader036.vdocument.in/reader036/viewer/2022062518/56649ee45503460f94bf371c/html5/thumbnails/15.jpg)
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
![Page 16: Factoring Polynomials. Factoring Numbers You will be learning in this lesson how to “undo” the process of multiplying polynomials that you learned to](https://reader036.vdocument.in/reader036/viewer/2022062518/56649ee45503460f94bf371c/html5/thumbnails/16.jpg)
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
![Page 17: Factoring Polynomials. Factoring Numbers You will be learning in this lesson how to “undo” the process of multiplying polynomials that you learned to](https://reader036.vdocument.in/reader036/viewer/2022062518/56649ee45503460f94bf371c/html5/thumbnails/17.jpg)
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
![Page 18: Factoring Polynomials. Factoring Numbers You will be learning in this lesson how to “undo” the process of multiplying polynomials that you learned to](https://reader036.vdocument.in/reader036/viewer/2022062518/56649ee45503460f94bf371c/html5/thumbnails/18.jpg)
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
![Page 19: Factoring Polynomials. Factoring Numbers You will be learning in this lesson how to “undo” the process of multiplying polynomials that you learned to](https://reader036.vdocument.in/reader036/viewer/2022062518/56649ee45503460f94bf371c/html5/thumbnails/19.jpg)
Guided Practice
Factor the greatest common factor (monomial) from each expression
a)
b)
c)
d)
e)
f)
![Page 20: Factoring Polynomials. Factoring Numbers You will be learning in this lesson how to “undo” the process of multiplying polynomials that you learned to](https://reader036.vdocument.in/reader036/viewer/2022062518/56649ee45503460f94bf371c/html5/thumbnails/20.jpg)
Guided Practice
Factor the greatest common factor (monomial) from each expression
a)
b)
c)
d)
e) not factorable (prime)
f)
![Page 21: Factoring Polynomials. Factoring Numbers You will be learning in this lesson how to “undo” the process of multiplying polynomials that you learned to](https://reader036.vdocument.in/reader036/viewer/2022062518/56649ee45503460f94bf371c/html5/thumbnails/21.jpg)
Practice 4
• Handout