Chapter 5 – 5-1 MonomialsMon., Oct. 19th
Essential Question:Essential Question:
•Can you apply basic arithmetic operations to polynomials, radical expressions andcomplex numbers
Target: Students will multiply and divide monomials
Agenda:
•Test
•Chapter 5 – 5-1 Monomials In-class
•Chapter 5 Homework: Pg. 226 - #19-41 odd
Chapter 5 – 5-1 MonomialsWed., Oct. 21st
Essential Question:Essential Question:
•Can you apply basic arithmetic operations to polynomials, radical expressions andcomplex numbers
Target: Students will identify the differencesbetween a monomial and polynomial and be able tosimplify polynomials
Agenda:
•Chapter 5-1 Homework: Pg. 226 - #19-41 odd
•5-2 Introduction
•5-2 Homework (Pg. 231 #17-33 odd, 37-49 odd)
Monomials
•An expression that is a number, a variable, or aproduct of a number and one or more variables
•Cannot contain variables in denominator, variableswith exponents that are negative, or variables underradicals
Monomials
Not Monomials
5b, -w, 23, x2, 1/2x3y4
1/n2, √x, x+8, a-1
Terms
•Constants:
•Monomials that contain no variable (i.e., 23, -1)
•Coefficient:
•Numeric factor of a monomial
•(4x – coefficient is 4)
Terms
•Degree of a monomial:
•Sum of the exponents of its variables
•4x2y3 - degree is 5
•Power:
•An expression of the form xn
Simplifying Expressions
•When exponents are multiplied – addtheir powers for the like variables:
(3x2y3)(2xy2) =
= 3 ∙ x ∙ x ∙ y ∙ y ∙ y ∙ 2 ∙ x ∙ y ∙ y (expanded form)
= 3 ∙ 2 ∙ x3 ∙ y5 (simplified)
For any real number a and integer m andn, am ∙ an = am+n
Simplifying Expressions
•When exponents are divided – subtracttheir powers for the like variables:
(3x2y3)/(2xy2) =
3 ∙ x ∙ x ∙ y ∙ y ∙ y (expanded form)
2 ∙ x ∙ y ∙ y
= 3xy/2 (simplified)
For any real number a and integer m and n,
am / an = am-n
Simplifying Expressions
•When exponents are to a power – multiplytheir powers for the like variables:
(3x2y3)2=
= 32 ∙ x2 ∙2 ∙ y3 ∙ 2
= 9x4y6
For any real number a and integer m and n,
(am)n = am ∙ n
Negative Exponent
•For any real number a ≠ 0 and anyinteger n, a-n = 1/an and 1/a-n = an
• therefore: a-4 = 1/a4
•And: 1/a-3 = a3
5.2 POLYNOMIALS
Polynomials
•Definition: Polynomial is a monomial or a sum ormonomials
•Monomials that make up polynomial are called TERMS ofthe polynomial
•X2, xy, x, each are different TERMS (Variable ANDexponents do not match – these CANNOT be combined)
•X2 , 4x2 , 21x2 are all the same TERM (Variable ANDexponent DO match – these CAN be combined)
Polynomials
•Variables:
•Terms of the unknown shown by a letter (x, y, ab, …)
•Coefficients:
•Numbers multiplied to variables
•Exponents:
•Repeated multiplication
•Leading Coefficient:
•The coefficient of the highest power
•Constant:
•The number with no variable
Polynomials
•Degree of Polynomial: Is the degree of the monomial withthe greatest degree
•5x3y5 – 9x4 - Degree is 8
•Standard Form:
•Write polynomial terms from highest to lowestexponents (x3 + x2 + 5)
Simplifying Polynomials
•To simplify polynomials means to perform the operations andcombine LIKE terms
•Add & Subtract – COMBINE like terms
•Multiply
•multiply coefficients
•add exponents to like variables
•Combine like terms
•Powers
•Raise coefficients to power
•Multiply exponents to like variables
•Combine like terms
Chapter 5 – Factoring BootcampFri., Oct. 23rd
Essential Question:Essential Question:
•Can you apply basic arithmetic operations to polynomials, radical expressions andcomplex numbers
Target: Students will simplify polynomial quotientsby factoring
Agenda:
•Pick up 5.1 homework
•5-2 Homework (Pg. 231 #17-33 odd, 37-49 odd)
•Quiz 5.1 & 5.2
•Factoring Bootcamp
•Factoring Homework - Worksheet
Factoring
•Finding Greatest Common Factor (GCF)
•Difference of 2’s
•Factoring Trinomials with leading coefficients of 1
•Factoring Trinomials with leading coefficients other than 1
•Factoring 4 terms by grouping
5.4 FACTORINGPOLYNOMIALS
Grouping, Trinomials,Binomials, GCF & SolvingEquations
Factoring a polynomial means expressing it asa product of other polynomials.
Factoring is a method to solve. To solve apolynomial, set it equal to zero.
Factor it.
Set each factor with a variable equal to zeroand solve.
This lesson focuses on methods to factor.
FACTORING POLYNOMIALS
1. Greatest Common Monomial Factor (GCF)
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2. Is it a Binomial?
a. Difference of Two Squares (DOS)
b. Difference of Two Cubes (DOC)
c. Sum of Two Cubes (SOC)
3. Is it a Trinomial?
a. Perfect Square Trinomial (PST)
b. Guess Method
4. Four or more terms?
a. Grouping
Factoring polynomials with a commonmonomial factor (using GCF).
**Always look for a GCF before usingany other factoring method.
Factoring Method #1
Step 1:
Step 2: Divide by GCF
The answer should look like this:
Factor these on your ownlooking for a GCF.
A “Difference of Squares” is a binomial(*2 terms only*) and it factors like this:
Some examples of variable squares are
Factoring Method #2
To factor a DOS, express each term asa square of a monomial then apply therule...
Here is another example:
Try these on your own:
Watch for GCFs and subsequent DOS.
Sum and Difference of Cubes:
Another binomial factoring technique
Note: the resulting quadratic is always prime and willprovide two imaginary solutions! Do not factor it!
Some examples of cubes are 1, 8, 27, 64, 125 and
Rewrite as cubes
Write each monomial as a cube andapply either of the rules.
Apply the rule for sum of cubes:
Rewrite as cubes
Apply the rule for difference of cubes:
Factoring Method #3
Factoring a trinomial in the formthat mimics:
But looks like this instead
Factors of +8: 1 & 8
2 & 4
-1 & -8
-2 & -4
Since we still have squares we must
See if we can still factor.
This is a Difference of squares!
Lets do another example:
Find a GCF
Factor trinomial
Always check for GCF before you do anything else.
Since neither is a DOS, You are Done!
You try
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You can check with FOIL
Factoring Technique #4
Factoring By Grouping for polynomials
with 4 or more terms
1. Group the first set of terms and last setof terms with parentheses.
2. Factor out the GCF from each group so thatboth sets of parentheses contain the samefactors.
3. Factor out the GCF again (the GCF is thefactor from step 2).
Step 1: Group
Example 1:
Step 2: Factor out GCF from each group
Step 3: Factor out GCF again
Example 2:
GCF
Split in
half
GCF each half
DOS
Try using factoring now to solve for x
Answers:
Factoring Completely
•Now that we’ve learned all the types offactoring, we need to remember to use them all.
•Whenever it says to factor, you must break downthe expression into the smallest possible
factors.
•Let’s review all the ways to factor.
Types of Factoring
1.Look for GCF first.
2.Count the number of terms:
a)4 terms – factor by grouping
b)3 terms -
•look for perfect square trinomial
•if not, factor using the BX term
c)2 terms -
look for difference of squares, sum ordifference of cubes
•If any ( ) still has an exponent of 2 or more, see if you canfactor again.
Chapter 5 – Dividing PolynomialsThurs., Oct. 29th
Essential Question:Essential Question:
•Can you apply basic arithmetic operations to polynomials, radical expressions and complexnumbers
Target: Students will divide polynomials using longdivision
Agenda:
•Pick up homework
•5-4 Factoring Homework
•(pg. 236 – 5, 12, 15-19 all, 33, 39-43 add)
•Finish 19-26 on Factoring Worksheet
•Quiz Factoring (CAN USE HOMEWORK)
•5-3 – Dividing using Long Division
•BRING HEADPHONES on MONDAY
Chapter 5 – Review – 5.1 – 5.4Nov. 4th
Essential Question:Essential Question:
•Can you apply basic arithmetic operations to polynomials, radical expressions and complexnumbers
Target: Students will review concepts learned in Chapter5
Agenda:
•5-3 – Day 2 Synthetic division Homework Review
•5.1 – 5.2 Quiz Review
•Factoring Quiz Review
•Move into Group for Review Work
•Get Review Packet checked off for NOTE SHEET approval