Download factor

Day 1
CLASSIFYING POLYNOMIALS
& ADDING AND SUBTRACTING
EXPRESSIONS
Explain the difference between:
Constant
variable
coefficient
Give an example for each term.
Parts of a Polynomial
P(x) = 2x3 – 5x2 – 2x + 5
Constant
variable
coefficient
Review: Simplify:
1.
8s – 9t + 4s – 3t
2.
4n2 – 7n2 + n2
3.
2b – 6 + 9b
Sam’s math teacher gave him a problem to simplify.
His work is below. Explain to Sam where he went
wrong and correct his mistakes for him.
Simplify:
4x2 + 10y – 12x2 + 2y
Sam’s answer:
16x4 + 12y2
9.1 Adding and Subtracting Polynomials
Defn: A Monomial is an expression that is a number,
a variable, or a product (multiplication only) of a
number and one or more variables.
Exs:
Degree of a Monomial
Is the SUM of the exponents of its VARIABLES
EX: What is the degree of the monomial:
1.
2.
3.
4.
Polynomial
Is a monomial or the SUM or DIFFERENCE of two or
more monomials
Ex:
Degree of a polynomial
Is the GREATEST exponent of the polynomial.
What is the degree of the polynomial:
Standard Form of a polynomial:
A polynomial in which the degrees of its monomial
terms DECREASE from LEFT to RIGHT
Classifying Polynomials
We can classify polynomials in two ways:
1) By the number of terms
# of Terms
Name
Example
1
Monomial
3x
2
Binomials
2x2 + 5
3
Trinomial
2x3 + 3x + 4
4
Polynomial with 4 2x3 – 4x2 + 5x + 4
terms
Classifying Polynomials
We can classify polynomials in two ways:
2) By the degree of the polynomial (or the largest
degree of any term of the polynomial.
Degree
Name
Example
0
Constant
7
1
Linear
2x + 5
2
Quadratic
2x2
3
Cubic
2x3 – 4x2 + 5x + 4
4
Quartic
x4 + 3x2
5
Quintic
3x5 – 3x + 7
Graphs are based on degrees!
Linear
Constant
Quadratic
Cubic
Quartic
Classifying Polynomials
Write each polynomial in standard form. Then
classify it by degree AND number of terms.
1.
-7x2 + 8x5
3. 4x + 3x + x2 + 5
2. x2 + 4x + 4x3 + 4
4. 5 – 3x
Complete the chart:
Polynomial
3x2 + 5x – 7
3x2
2x + 5
-123
-4x
6
3x3 + 2x2 – 1
x3 - 4x2
2x3
3x2 – 4
Degree Name of Degree? Name by # of
terms
Recall:
Simplify:
1.
5x + 6y + 8x – 3y + 9

All you are doing is combining like
terms!!!
NEVER change the exponent when
adding or subtracting!!!

Simplify:
Ex 1:
Ex 2:
Ex. 3
Ex. 4
*** When subtracting:
ALWAYS DISTRIBUTE THE NEGATIVE then ADD!
Ex. 5
Ex. 6
Error Analysis: Describe and correct the error
in finding the difference of the polynomials.
Line 1: (4x2 – x + 3) – (3x2 – 8x – 9)
Line 2: 4x2 – x + 3 – 3x2 – 8x – 9
2
2
Line 3: 4x – 3x – x – 8x + 3 – 9
2
Line 4: x – 9x – 6
Day 2/3
MULTIPLYING LINEAR EXPRESSIONS
Warm Up
Homework 6.1
Multiplying Monomials
Multiply Coefficients
&
Add Exponents
1. (4x2)(5x)
2. (-2xy)(6x4y)
3. (-3w)(-4w)
4. (5yz3)(xyz)
Multiplying Monomials and Polynomials
When you are multiplying a Monomial with a
Polynomial you DISTRIBUTE!
3x(-2x + 3)
Multiply
-4x(x – 7)
Multiply
5x2(2x + 1)
Multiply
x(3x2 + 4x + 5)
Multiplying Binomials and Polynomials
There are 2 methods:
Box Method
Distribution
Multiplying Binomials and Polynomials
There are 2 methods:
Box Method
(2x – 1)(4x + 5)
Multiplying Binomials and Polynomials
There are 2 methods:
Distribution
(2x – 1)(4x + 5)
Multiply
(x + 2)(-3x – 2)
Multiply
(3x + 4)(5x – 6)
Multiply
(x + y)(7x – 4y)
Multiply
(3a – b)(4a – 8b)
Multiply
(x + 2)(x2 + 5x – 3)
Multiply
(4x – 5)(x2 – 7x – 2)
Special Cases
A) The Square of a Binomial:
(a + b)2 = a2 + 2ab + b2
(a – b)2 = a2 – 2ab + b2
EX: (x + 3)2
Special Cases
B) The Difference of Squares
(a + b)(a – b) = a2 – b2
EX: (x – 5)(x+5)
Area and Perimeter
A Rectangle has a length of 3x – 5 and a
width of 2x + 6
Find the Area
Find the Perimeter
Find the area of the shaded region
Graphing
Quadratic function
A QUADRATIC FUNCTION is a function that
can be written in the standard form:
f(x) = ax2 + bx + c where a≠ 0
Graphing quadratic
The graph of a quadratic function is Ushaped and it is called a PARABOLA.
Parts of a parabola
VERTEX: the highest
or lowest point on the
graph.
Max if open down
Min if open up
Parts of a parabola
AXIS OF
SYMMETRY:
vertical line (x = a)
that divides the
parabola into two
symmetrical parts.
Ex 3:
Vertex: ____
2. Axis of symmetry: _____
3. X-intercept:_____
4. Y-intercept:_____
1.
Try Some!
Identify the vertex, axis of symmetry, x
intercept and y intercept.
Try Some!
Identify the vertex, axis of symmetry, x
intercept and y intercept.
Complete the investigation:
Pairs will display their work under the doc cam.
Make sure your work is neat!!!
Graphing in a Calculator
Properties of Parabolas
Graph the following functions on a graphing
Calculator. What do you notice?
1. y = x2
2. y = -x2
3. y = ½x2 + 2
4. y = -½x2 + 2
Properties of Parabolas
If a is POSITIVE then the parabola
opens UP
IF a is NEGATIVE then the parabola
opens DOWN
Properties of Parabolas
Graph the following functions on a graphing
Calculator. What do you notice?
1. y = x2 + 3x – 5
2. y = -x2 + 2x – 5
3. y = 3x2 + 2x + 3
4. y = -3x2 + 4x + 3
Graphing y=ax2 + bx + c
- Y –intercept is (0,c)
Calculator Commands
Graph: Y=  Graph
Vertex: 2nd Trace  Min or Max
(left bound, right bound, enter)
X-Intercepts: 2nd Trace  Zero
(find each one separately)
Y-Intercept: 2nd Graph  find where x is zero
Use a graphing calculator to answer the following
questions. All equations must be in y= before we
can graph them using the calculator.
Ex. 1
1.
2.
3.
4.
5.
6.
Standard Form:__________
Graph:
Vertex: ____
Axis of symmetry: _____
X-intercept:_____
Y-intercept:_____
y
x
Ex 2:
1.
2.
3.
4.
5.
6.
Standard Form:______________
Graph:
Vertex: ____
Axis of symmetry: _____
X-intercept:_____
Y-intercept:_____
y
x
Ex 3:
1.
2.
3.
4.
5.
6.
Standard Form:______________
Graph:
Vertex: ____
Axis of symmetry: _____
X-intercept:_____
Y-intercept:_____
y
x
1.
Suppose an eagle is flying 30 feet above a canyon when
it drops a stick from its claws. The height, h, of the
stick t seconds after it is released can be modeled by
the function h=-16t2 + 30.
A. How high above the ground is the stick after 1
second?
B. When will the stick hit the ground?
(The height when an object hits the ground is
0.)
1C
Look at the graph. Why is only the first
quadrant shown?
2.
A monkey drops an orange from a branch26 ft above the
ground. The force of gravity calls the orange to fall to
earth. The function h = -16t2 + 26 gives the height of the
orange h in feet after t seconds.
When will the orange hit the ground?
(The height when an object hits the ground is 0.)
3.
Use a graph to find the minimum/maximum points
(vertex)!
A. The total profit made by an engineering firm is
given by the function p = x2 – 25x + 5000. Find the
minimum profit.
Quadratic Equations
SOLVING BY GRAPHING
Quadratic Equation
Standard Form of Quadratic Function:
y = ax2 + bx + c
Standard Form of Quadratic Equation:
0 = ax2 + bx + c
Solutions
A SOLUTION to a quadratic equation is
a value for x, that will make 0 = ax2 + bx
+ c true.
A quadratic equation always have 2
solutions.
5 ways to solve
There are 2 ways to solve quadratic equations:
Factoring
Graphing
Solving by Graphing
For a quadratic function, y = ax2 +bx + c,
a zero of the function, or where a
function crosses the x-axis, is a solution
of the equations ax2 + bx + c = 0
Examples
Solve x2 – 5x + 2 = 0
Examples
Solve x2 + 6x + 4 = 0
Examples
Solve 3x2 + 5x – 12 = 8
Examples
Solve x2 = -2x + 7
Factoring
Factors
What are the following expressions factors of?
1. 4 and 5?
2. 5 and (x + 10)
3. 4 and (2x + 3)
4. (x + 3) and (x - 4)
5. (x + 2) and (x + 4)
6. (x – 4) and (x – 5)
Factors
Factors are numbers or expressions that you
multiply to get another number or expression.
Ex. 3 and 4 are factors of 12 because 3x4 = 12
Greatest Common Factor (GCF)
 A greatest common factor is the biggest number that
divides evenly into 2 or more terms.
 The GCF of variables with exponents is the common
variable with the smallest exponent
Ex) What is the GCF of 3x4 + 6x2
What is the GCF of
25a2 and 15a?
5a
Let’s go one step further…
1) FACTOR 25a2 + 15a.
Find the GCF and divide each term
5a + ___
3 )
25a2 + 15a = 5a( ___
25a 2
5a
15a
5a
Check your answer by distributing.
2) Factor 18x2 - 12x3.
Find the GCF
6x2
Divide each term by the GCF
3 - ___
2x )
18x2 - 12x3 = 6x2( ___
18 x 2
6x2
12 x 3
6 x2
Check your answer by distributing.
3) Factor 28a2b + 56abc2.
GCF = 28ab
Divide each term by the GCF
2
2
2
a
2c
28a b + 56abc = 28ab ( ___ + ___ )
28a 2b
28ab
56abc 2
28ab
Check your answer by distributing.
28ab(a + 2c2)
4) Factor 28a2 + 21b - 35b2c2
GCF = 7
Divide each term by the GCF
28a2 + 21b - 35b2c2 = 7 ( ___
4a2 + ___
3b - ____
5b2c2 )
28a 2
7
21b
7
35b 2 c 2
7
Check your answer by distributing.
7(4a2 + 3b – 5b2c2)
5) Factor 20x2 - 24xy
1. x(20 – 24y)
2. 2x(10x – 12y)
3. 4(5x2 – 6xy)
4. 4x(5x – 6y)
6) Factor 16xy2 - 24y2z + 40y2
1. 2y2(8x – 12z + 20)
2. 4y2(4x – 6z + 10)
3. 8y2(2x - 3z + 5)
4. 8xy2z(2 – 3 + 5)
GCF
EX: 4x2 + 20x – 12
EX: 9n2 – 24n
Try Some!
Factor:
a. 9x2 +3x – 18
b. 7p2 + 21
c. 4w2 + 2w
Factoring when a=1
Factors of Quadratic Expressions
When you multiply 2 binomials:
(x + a)(x + b) = x2 + (a +b)x + (ab)
This only works when the coefficient for
x2 is 1.
Finding Factors of Quadratic Expressions
When a = 1:
x2 + bx + c
Step 1. Determine the signs of the factors
Step 2. Find 2 numbers that’s product is c,
and who’s sum is b.
Sign table!
2nd
sign
+
Same
Different
Question
1st
sign
Answer
+
-
(x+ )(x+ ) (x - )(x- )
+ or (x + )(x - )
OR
(x - )(x + )
Examples
Factor:
1. X2 + 5x + 6
Examples
Factor:
2. x2 – 10x + 25
Examples
Factor:
3. x2 – 6x – 16
Examples
Factor:
4. x2 + 4x – 45
Examples
Factor:
1. X2 + 6x + 9
2. x2 – 13x + 42
3. x2 – 5x – 66
4. x2 – 16
Factoring when a≠1
More Factoring!
When a does NOT equal 1.
Steps
1. Slide
2. Factor
3. Divide
4. Reduce
5. Slide
Example!
Factor:
1.
3x2 – 16x + 5
Example!
Factor:
2.
2x2 + 11x + 12
Example!
Factor:
3.
2x2 + 7x – 9
Try Some!
Factor
1. 5t2 + 28t + 32
2. 2m2 – 11m + 15
Quadratic Equations
2 ways to solve
There are 2 ways to solve quadratic equations:
Factoring
Graphing
SOLVING BY FACTORING
Factoring
Solve by factoring;
x2 + 7x = 18
Factoring
Solve by factoring;
2x2 – 11x = -15
Factoring
Solve by factoring;
1. 2x2 + 4x = 6
3. x2 – 9x + 18 = 0
2. 16x2 – 8x = 0