Algebra 2
Name________________________
Hour_______
Guided Notes
Unit 3
Polynomials, solving, and Graphing
Quadratics
“Life has no remote. Get up and change it yourself.”
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Algebra 2
Algebra 2 Unit 3
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Learning Goal: Students will factor polynomials and solve quadratic equations using
various methods including factoring, completing the square, and the quadratic formula, as
well as graph quadratic functions that are represent by all three formats—vertex form,
intercept form, and general form.
0 1 2 3 4
Objective in words
Students will be able to identify the degree and leading coefficient of a polynomials as
well as name the polynomial.
Students will be able to factor basic polynomials.
Students will be able to factor more complex polynomials.
Students will be able to solve quadratic equations by factoring.
Students will be able to solve quadratic equations by completing the square.
Students will be able to solve quadratic equations using the quadratic formula.
Students will be able to convert between different forms of a quadratic function—
vertex, general, and intercept.
Students will be able to graph quadratic functions in vertex form and identify the
vertex and axis of symmetry.
Students will be able to graph quadratic functions in general form and identify the
vertex and axis of symmetry.
Students will be able to graph quadratic functions in intercept for and identify the
vertex and axis of symmetry.
Score 4
Score 3
Score 2
Score 1
Score 0
I can accurately solve problems related to quadratic functions, including
factoring and graphing, and identify key properties and procedures for
doing so, never making a mistake. I can also apply quadratics to application
and real world problems.
I can accurately solve problems related to quadratic functions, including
factoring and graphing, and identify key properties and procedures for
doing so, rarely making a mistake.
I understand and can solve basic quadratic problems, but have trouble
identifying and remembering all of the procedures and properties related to
quadratics.
With help, I can solve these problems as long as they are not too complex.
Even if I have help, I cannot do these problems….YET!
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Algebra 2
3.1 Polynomials and Basic Factoring
VOCABULARY
DEFINITION
EXAMPLE
Monomial
A number, variable or the
product of a number and
variable(s) with whole number
exponents.
10
3x
-1.8ab2
10 degree: zero
3x degree: one
-1.8ab2 degree: three
Degree of a
Monomial
The sum of the exponents of the
variables in the monomial.
Polynomial
A sum of monomials, each is
called a term.
2x3+x2+7
Degree of a
Polynomial
The greatest degree of its
monomial terms.
2x3+x2+7
degree: three
Leading Coefficient
When a polynomial is written so that the
exponents of a variable decrease from left to
right, the coefficient of the first tem is called the
leading coefficient.
2x3+x2+7
Leading coefficient: 2
POLYNOMIAL
Monomial
Binomial
Trinomial
1) Fill in the table:
Expression
Is it a polynomial?
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Y or N
Classify by degree and number of terms
Degree:
# of terms:
2x2 + x - 5
Y or N
Degree:
# of terms:
6n4 - 8n
Y or N
Degree:
# of terms:
n-2 + 3
Y or N
Degree:
# of terms:
7bc3 + 4b4c
Y or N
Degree:
# of terms:
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Algebra 2
Rearrange the polynomial so the exponents are in descending order (standard form).
Then identify the degree and the leading coefficient.
2) 5𝑦 − 2𝑦 4 + 9
3) 15𝑥 + 3 − 𝑥 3
4) 3𝑥 2 𝑦 3 + 2𝑥 − 7𝑥𝑦 3
THE FIRST STEP IN FACTORING SHOULD ALWAYS BE TO CHECK TO SEE
IF THERE IS A GREATEST COMMON MONOMIAL FACTOR THAT CAN
BE FACTORED OUT!!!
Factor out the greatest common monomial.
5) 12𝑥 + 42𝑦
6) 4𝑥 4 + 24𝑥 3
7) 7𝑤 5 − 35𝑤 2 + 𝑤
Factor the binomials.
Difference of Two Squares Pattern

a2 – b2 =
Example_____________________________________
8) 𝑦 2 − 16
9) 25𝑚2 − 36
10) 4𝑦 2 − 64
11) 𝑥 2 − 49𝑦 2
12) 𝑥 2 − 81𝑦 2
13) 64𝑐 2 − 16
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Algebra 2
14) 400 − 49𝑣 2
15) 8𝑥 2 − 162
Factor the following trinomials.
16) 𝑥 2 + 11𝑥 + 18
17) 𝑥 2 − 5𝑥 + 6
18) 𝑦 2 + 2𝑦 − 15
19) 𝑚2 + 𝑚 − 20
20) 𝑡 2 + 9𝑡 + 14
21) 𝑤 2 + 18𝑤 + 56
Summary
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Algebra 2
3.2 Factoring ax2 + bx + c
Factor the following trinomials.
1) 4𝑥 2 + 16𝑥 + 15
 Step 1: Find ac and b.
ac=____________
b=____________
 Step 2: Find two numbers that multiply to ac and add to b.
_______ x ________= ac
________+________=b

Step 3: Split the middle bx term using the two numbers found in step 2.

Step 4: Factor by grouping the first two terms and the second two terms, taking
out the greatest common monomial factor from each.
*Check to see what is in the parentheses matches.
Answer ____________________________
Try these:
2) 2𝑥 2 − 7𝑥 + 3
3) 3𝑛2 + 14𝑛 − 5
4) −4𝑥 2 + 12𝑥 + 7
5) −𝑥 2 + 𝑥 + 20
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Algebra 2
6) −2𝑦 2 − 5𝑦 − 3
7) 3𝑥 2 − 𝑥 − 2
Notice that this method still works when a=1.
TRY:
8) 𝑛2 − 6𝑛 + 8
Factor completely.
First, __________________________________________________________.
9) 4𝑥 3 − 44𝑥 2 + 96𝑥
10) 50ℎ4 − 2ℎ2
11) 3𝑥 2 − 12𝑥
12) 2𝑦 3 − 12𝑦 2 + 18𝑦
13) 2𝑎4 + 21𝑎3 + 49𝑎2
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Algebra 2
14) You have a metal plate that you have drilled a hole into. The entire area enclosed by
the metal plate is given by 5𝑥 2 + 12𝑥 + 10 and the area of the hole is given by 𝑥 2 + 2.
Write an expression for the area in factored form of the plate that is left after the hole is
drilled.
Summary

3.3 Forms of a Quadratic Equation and Solving
Equations
Zero Product property _______________________________________________
____________________________________________________________________
 Solutions to a quadratic equations can be called ___________________ or
________________ or__________________
3 Forms of a Quadratic Equation
General Form
Vertex Form
Intercept Form
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Algebra 2
State the zeros/roots of the graph.
Find the zeros of the equations in Intercept Form (Already factored).
1) 𝑦 = (𝑥 − 4)(𝑥 + 2)
2) 𝑦 = 5(𝑥 − 5)(𝑥 − 1)
3) 𝑦 = 2𝑥(𝑥 + 7)(3𝑥 − 2)
Find the zeros of the equations in Vertex Form.
You solve these by taking square roots.
Remember when you solve an equation by taking a square root you need_________.
This is because if you think about 𝑥 2 = 4, there are two possible solutions for x, ___ & ___
4) 𝑦 = 2(𝑥 − 3)2 − 18
5) 𝑦 = 3𝑥 2 − 48
6) 𝑦 = −2(𝑥 + 4)2 + 10
Solve equations in General Form.
When solving these, don’t forget:

You need to make sure the equation equals _______________.

You need to _______________________________ if it is in general form.
Solve the following.
7) 2𝑥 2 + 8𝑥 = 0
8) 5𝑛2 = 15𝑛
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Algebra 2
9) 3𝑠 2 − 9𝑠 = 0
10) 4𝑠 2 = −14𝑠
11) 𝑛2 − 7𝑛 − 30 = 0
12) 𝑥 2 = 9𝑥 − 20
13) 𝑥 2 − 11𝑥 + 24 = 0
14) 𝑦 2 + 3𝑥 = 18
Find the zeros or solve.
15) 𝑦 = 3𝑥 2 − 15𝑥 + 18
Solve or find the zeros.
17) 3𝑛2 + 13𝑛 + 4 = 0
16) 𝑦 = 2𝑥 2 + 7𝑥 − 15
18) 𝑦 = 2𝑥 2 − 7𝑥 + 3
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Algebra 2
Solve or find the zeros/roots of the following equations.
19) 3𝑥 3 + 18𝑥 2 = −24𝑥
20) 𝑦 = 8𝑥 2 + 36𝑥 + 16
21) 𝑐 4 − 64𝑐 2 = 0
22) z 3 = 4z
23) 3𝑛2 = 40 + 19𝑛
24) 2𝑥 2 − 12𝑥 − 18 = −4
25) −2𝑟 2 − 11𝑟 − 18 = −4𝑟 2 − 3𝑟 + 6
26) −11 − 7𝑘 = −7𝑘 − 𝑘 2 − 7
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Algebra 2
Application Problems:
27) A ball is tossed into the air from a height of 8 feet with an initial velocity of 8 feet per
second. Find the time t (in seconds) it take for the object to reach the ground by solving
the equation −16𝑡 2 + 8𝑡 + 8 = 0
28) Students found that by launching a water ballooons 42 feet per second at a precise
angle, they sould hit the statue as shown below. Here’s the equation they worked out to
represent the path of the balloons: H(t) = -3t2 + 42t + 58. The t represents the number
of seconds and H is the height of the balloon in feet.
a) If the statue is 13 feet tall, how many seconds did it take for the balloon to reach
the statue after the launch?
Summary
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Algebra 2
3.4 Completing the Square to Solve Quadratics
Vocabulary:

To complete the square, you wish to turn the expression into a
perfect square trinomial (Like 𝑥 2 − 20𝑥 + 100 because this factors
into (𝑥 − 10)2)
o To complete the square for the expression x2 + bx, add
________.
Because…..
Finding c:
Find the value of c that makes the expression a perfect square trinomial. Then
write the expression as the square of a binomial.
1. x2 + 6x + c
2. x2 – 24x + c
Solve by Completing the Square:
Solve the equation by completing the square. Notice, YOU CANNOT factor these
like we did in earlier solving problems.
o Step 1: Move over the constant “c” term if needed.
𝑏 2
o Step 2: Find (2) and add to each side.
o Step 3: Write the expression as the square of a binomial.
o Step 4: Solve the equation using the square root method.
3. x2 + 4x = 10
4. x2 + 8x + 1 = 0
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Algebra 2
5. x2 – 10x +13 = 0
Solve by Completing the Square with a other than 1:
6. 3x2 + 6x = 72
Old Method:
7.8𝑥 2 + 16𝑥 = 42
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Algebra 2
Rewriting the forms of a quadratic:
8. Write y = x2 – 10x + 22 in vertex form.
9. The height, h, in feet of a baseball t seconds after it is hit is given by: h = -16t2
+ 96t +3. Rewrite in vertex form.
a) What is the vertex (maximum height of ball)?
10. Given 𝑦 = 2(𝑥 − 1)2 + 7, rewrite into general form.
Summary
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Algebra 2
3.5 Using the Quadratic Formula Solve Quadratics
Vocabulary:

The quadratic formula is _______________________________
IF ALL ELSE FAILS, THE QUADRATIC FORMULA CAN ALWAYS BE
USED TO SOLVE A QUADRATIC EQUATION.
Solve the following quadratic equations using the quadratic formula.
 Step 1: Set equation equal to zero
 Step 2: Identify a, b, and c.
 Step 3: Plug a, b. and c into the quadratic formula
 Step 4: Simplify to get answer(s)
1) 3𝑥 2 + 5𝑥 = 8
2) 𝑥 2 − 8 = −7𝑥
3) 2𝑥 2 − 𝑥 − 7 = 0
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Algebra 2
4) 𝑥 2 + 7𝑥 + 5 = 0
Solve the following quadratic using all three methods below:
𝑥 2 − 2𝑥 − 3 = 0
Factoring:
Completing the Square:
Quadratic formula:
Summary
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Algebra 2
3.6 Graphing in Vertex Form
VERTEX FORM
Quadratic Functions: Parabolas
(Identify Key Characteristics)
Minimum or Maximum? The value of a decides if it is max or min.

If a>0, then the graph opens _____________ and has a ____________

If a<0, then the graph opens _____________ and has a ____________
Graphing with Vertex Form:
Graph the Function. Identify the domain and range.
1. y = 3(x – 4)2 - 3
Steps to graph in vertex form.
1. Identify the constants:
a=
h=
k=
2. Plot (h, k) and draw in A of S.
3. Then evaluate the function for two other x values
near the A of S and reflect to other side of A of S.
**Remember: You need 5 points, vertex and two on
each side
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Algebra 2
2. Graph y = -(x – 1)2 + 5
𝟏
3. Graph y = -𝟐(x + 3)2
Writing an Equation for a Graph
4. Write an equation for this graph in vertex form.
5. Write an equation for this graph in vertex form.
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Algebra 2
6. Write the general form of the equation into vertex form. Recall to do this we
____________ _______ __________.
𝑦 = 2𝑥 2 + 12𝑥 − 4
Summary
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Algebra 2
3.7 Graphing in General Form
GENERAL FORM
Graphing with General Form:
Graph the function. Identify the domain and range.
Steps to graph standard quadratic functions.
1. y = 5x2 + 1
1. Identify the coefficients:
a=
b=
c=
2. Find the axis of symmetry (A of S) using:
b
x
2a
3. Draw the axis of symmetry: x =
4. Find the vertex plugging in the x from step 3 to
find y.
5. Identify the y-intercept: (0, c) and reflect this
point if possible.
6. Evaluate the function for other value(s) of x near
A of S. Plot these points, and reflect it to other side
of A of S.
**Remember: You need 5 points, vertex and two on
each side
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Algebra 2
2. y = 3x2
4. Graph y = -x2 – 5
3. y = x2 + 2x + 1
5. Graph y = 2x2 + 12x + 9
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Algebra 2
6) A rocket is launched from 180 feet above the ground at time t=0. The function
that models this situation is given by h(t)= - 16t2 + 96t + 180, where t is measures
in seconds and h is the height above the ground measured in feet.
a) Determine the height of the rocket two seconds after it was launched.
b) Determine the time the rocket reaches its maximum height and the
maximum height obtained by the rocket. (Hint: Think about the vertex)
Summary
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Algebra 2
3.8 Graphing in Intercept Form
INTERCEPT FORM
Graphing with Intercept Form:
Graph the function. Identify the domain and range.
1. y = (x + 2)(x – 3)
Steps to graph in intercept form.
1. Identify the x-intercepts:
p=
q=
2. Find the A of S using:
pq
x
2
3. Plot the vertex using x above and plugging in the
x to find y.
4. Evaluate the function at one other x-value and
reflect this point using the A of S.
**Remember: You need 5 points, vertex and two on
each side
2. Graph y = 2(x + 2)(x –2)
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Algebra 2
3. Graph y = -(x + 1)(x – 5)
4. Graph y = (x – 3)(x – 7)
5. Take the above equation in #3, y = -(x + 1)(x – 5), and rewrite it in general form.
a) Show using 𝑥 =
−𝑏
2𝑎
for the general form above, that you get the same A of S.
Summary
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