Algebra I
Chapter 8/9 Notes Part 1
Section 8-1: Adding and Subtracting
Polynomials, Day 1
Polynomial –
Binomial –
Trinomial –
Degree of a monomial –
Degree of a polynomial –
Section 8-1: Adding and Subtracting
Polynomials, Day 1
Polynomial – a monomial or the sum of monomials (also
called terms)
Binomial – a polynomial with 2 terms
Trinomial – a polynomial with 3 terms
Degree of a monomial – the sum of the exponents of all
its variables
Degree of a polynomial – the greatest degree of any term
in the polynomial
Section 8-1: Adding and Subtracting
Polynomials, Day 1
Degree
0
1
2
3
4
5
6 or more
Name
Section 8-1: Adding and Subtracting
Polynomials, Day 1
Fill in the table
Expression
Polynomial?
Degree
Monomial,
Binomial, or
Trinomial?
Section 8-1: Adding and Subtracting
Polynomials, Day 1
Standard Form –
Leading Coefficient –
Ex) Write each polynomial in standard form.
Identify the leading coefficient.
a) 3x2 + 4x5 - 7x b) 5y- 9 - 2y4 - 6y3
Section 8-1: Adding and Subtracting
Polynomials, Day 1
Standard Form – the terms are in order from
greatest to least degree
Leading Coefficient – the coefficient of the first
term when written in standard form
Ex) Write each polynomial in standard form.
Identify the leading coefficient.
a) 3x2 + 4x5 - 7x b) 5y- 9 - 2y4 - 6y3
Section 8-1: Adding and Subtracting
Polynomials, Day 2
Find each sum
2
2
1) (2x + 5x- 7)+ (3- 4x + 6x)
2) (3y+ y3 - 5)+ (4y2 - 4y+ 2y3 +8)
Section 8-1: Adding and Subtracting
Polynomials, Day 2
Subtract the following polynomials
1) (3- 2x+ 2x2 )- (4x- 5+ 3x2 )
2) (7p+ 4p3 -8)- (3p2 + 2 - 9 p)
Section 8-2: Multiplying polynomial by
a monomial
Multiply
1) -3x2 (7x2 - x+ 4)
3)
2) 5a2 (-4a2 + 2a- 7)
2 p(-4p2 + 5p) - 5(2 p2 + 20)4) 15t(10y3t 5 + 5y2t) - 2y(yt 2 + 4y2 )
Section 8-2: Multiplying polynomial by
a monomial
Solve the equation. Distribute and combine like
terms first!
1) 5(4z+ 6) - 2(z- 4) = 7z(z+ 4) - z(7z- 2) - 54
Section 8-3: Multiplying Polynomials,
The Box Method
Steps for using the box
method:
1) Draw a box with
dimensions based on the
number of terms in the
polynomials
2) Fill in the box using
multiplication
3) Re-write the entire
answer as one polynomial
(combine any like terms)
Ex)
(x – 2)(3x + 4)
Section 8-3: Multiplying Polynomials,
The Box Method
Multiply
1) (2y – 7)(3y + 5)
2) (6x+ 5)(2x - 3x- 5)
2
Section 8-3: Multiplying Polynomials,
The Box Method
3) (m2 - 5m+ 4)(m2 + 7m- 3)
4) (t - 4)[(t 2 + 3t -8) - (t 2 - 2t + 6)]
Section 8-4: Special Products
Square of a sum – (a+ b)2  (a+ b)2 = (a+ b)(a+ b)
Find the product
2
1) (3x+ 5)
2) (2x - 5)2
Section 8-4: Special Products
Product of a Sum and Difference: (a + b)(a – b)
Multiply
1) (x + 3)(x – 3)
2) (6y – 7)(6y + 7)
Section 9-1: Graphing Quadratic
Functions, Day 1
Quadratic Function –
Parabola –
Axis of Symmetry –
Vertex (min/max) -
Section 9-1: Graphing Quadratic
Functions, Day 1
Quadratic Function – non-linear functions that can
written in the form, ax2 + bx+ c, where a cannot be
zero
Parabola – the shape of the graph of a quadratic. A ‘U’
shape either opening up or down
Axis of Symmetry – the vertical line that cuts a
parabola in half
Vertex (min/max) – the lowest or highest point on a
parabola
Section 9-1: Graphing Quadratic
Functions, Day 1
Section 9-1: Graphing Quadratic
Functions, Day 1
Fill in the table and graph the quadratic
equation y = 3x2 + 6x - 4
X
1
0
-1
-2
-3
Y
Section 9-1: Graphing Quadratic
Functions, Day 1
Find the vertex, axis of symmetry, and yintercept of each graph
1)
2)
Section 9-1: Graphing Quadratic
Functions, Day 1
Find the vertex, the axis of symmetry, and the yintercept of each function.
a) y = 2x2 + 4x- 3
b) y = -x2 + 6x+ 4
Section 9-1: Graphing Quadratic
Functions, Day 2
Section 9-1: Graphing Quadratic
Functions, Day 2
For each function, determine if the function has a min
or a max, find what that value is, then state the domain
and range.
1) f (x) = -2x - 4x+ 6
2
2
f
(x)
=
-x
+ 4x - 3
2)
Section 9-1: Graphing Quadratic
Functions, Day 2
Steps for graphing quadratics
(3 points MINIMUM!)
1st point) Find and plot the
vertex
2nd point) Find and plot the yintercept***
3rd point ) Mirror the yintercept across the axis of
symmetry and plot the 3rd
point
***If the y-intercept and the
vertex are the same, you must
choose a different 2nd point
Graph f (x) = x2 + 4x+ 3
Section 9-1: Graphing Quadratic
Functions, Day 2
Graph f (x) = 3x2 - 6x+ 2
(Plot 3 points!)
Section 9-1: Graphing Quadratic
Functions, Day 2
Linear, Exponential, and Quadratic Functions!
Linear Functions
Exponential
Functions
Quadratic Functions
As x inc., y dec.
Or
As x inc., y dec.
As x inc., y inc.
Or
As x inc., y dec.
As x inc., y inc. then
dec. OR
As x inc., y dec., then
inc.
Equation
Degree
Graph name
What does the
graph look like?
End behavior
Section 9-5: The Quadratic Formula,
Day 1
The Quadratic Formula:
The solutions of a quadratic equation ax2 + bx+ c = 0
Where a does not equal zero are given by the
following:
-b± b - (4ac)
x=
2a
2
Section 9-5: The Quadratic Formula,
Day 1
Steps for using the
quadratic formula:
1) Set the equation = 0
2) Label a, b, and c
3) Plug a, b, c into the
formula
4) Under Radical
5) Square Root
6) Split into 2
7) Simplify the 2 fractions
Solve using Q.F.
x -12x = -20
2
Section 9-5: The Quadratic Formula,
Day 1
Solve using Q.F. Round to
1) 3x2 + 5x-12 = 0
Nearest hundredth
2) 10x2 - 5x = 25
Section 9-5: The Quadratic Formula,
Day 2
Solve using Q.F.
1) 2x2 - 5x = -7
2) x2 +10x+ 25 = 0
Section 9-5: The Quadratic Formula,
Day 2
Discriminant –
Discriminant
Graph
Number of
Solutions
Section 9-5: The Quadratic Formula,
Day 2
Discriminant – a value found by taking b2 - 4ac that
determines the number of solutions
Discriminant
Positive
Zero
Negative
Two
One
None
Graph
Number of
Solutions
Section 9-5: The Quadratic Formula,
Day 2
Use the discriminant to determine how many
solutions the equation has. DO NOT SOLVE!
2
2
1) 4x + 5x = -3
2) 2x +11x+15 = 0
3) 9x2 - 30x+ 25 = 0