Download Polynomials and Basic Quadratics

Algebra 1
Vocabulary- Polynomial and Quadratics Unit
Name_____________________________________
Date _____________________________________
Objective:
To know the definitions of the key vocabulary words/terms/phrases in the Functions Unit. To
understand the difference between an object and a process as they apply to the Functions Unit.
In General
Object
As it applies to mathematical concepts
In General
Process
As it applies to mathematical concepts
1. Classify each of the following words as an object and/or the result of a process.
2. What should be the expectation of the term as an object and/or the result of a process.
Example: Root of a quadratic:
Noun- The points at which a function crosses the x-axis
Verb- The solution of a quadratic equation when set equal to 0.
Graph
Axes
x-intercept
y-intercept
Coordinates
Abscissa
1
Ordinate
Independent variable
Dependent variable
Slope
Symmetry
Axis of Symmetry
Operation
Parabola
Vertex
Polynomial
2
Algebra 1
Lesson- Polynomial Operations- Addition & Subtraction
Name:____________________________________
Date:_____________________________________
Objective:
1)
2)
To review properties of monomials and polynomials
To review the rules and processes for adding and subtracting monomials and polynomials
DO NOW:
Find the sum of 13, 25, and 32 without the use of a calculator
__________________________________________________________________________________________
What is an operation?
What is a monomial?
What is a polynomial?
What is the degree of a monomial?
Situation 1: Monomial and Monomial (M&M)
Rules:
Examples:
3m 2  17m 2
1.
2.
9 z 3  12 z 3
3
Situation 2: Monomial and Binomial
Rules:
Examples:
3x  (5 x  2)
3.
4.
2a  (7a  1)
Situation 3: Binomial and Binomial
Rules:
Examples:
5. (3x 2  2)  (4 x  7)
6. (14 x 2  7)  ( x  7)
Situation 4: Polynomial and Polynomial
Rules:
Examples:
7. (3x 2  2 x  1)  (4 x  7)
8. (14 x 2  7 x)  ( x 2  7 x  6)
4
Algebra 1
Lesson- Polynomial Ops- Multiplication & Division
Name:____________________________________
Date:_____________________________________
Objective:
To review the rules and processes for multiplying and dividing monomials and polynomials
DO NOW:
Find the difference of 25 and 13 without the use of a calculator
__________________________________________________________________________________________
Situation 1: Monomial and Monomial (M&M)
Rules:
Examples:
4m 2  6m
1.
2.
4m 2  6m
4.
(2 x 2  4)  (2 x)
6.
( x  4)( x  4)
Situation 2: Monomial and Binomial
Rules:
Examples:
4 f  (3a  2)
3.
Situation 3: Multiplying Binomial by Binomial
Rules:
Examples:
(2 x  1)( x  2)
5.
7.
(2 x  1)( x 2  2 x  1)
5
Algebra 1
WKST- Polynomial Operations- Mixed
Name:____________________________________
Date:_____________________________________
Perform the indicated operations:
1. 2 x 5  7 x 5
2. 16 x 3  7 x 3
5. 2 x 5  7 x 2
6. 21x 3  7 x 2
9. 4 x 2  (5 x 2  1)
10. 21x 3  (7 x  5x 3 )
3. 5 x10  7 x10
7. 12 x 5  6 x 5
11. 2 x 5 (3  x 2 )
4. 14 x 2  7 x 3
8. 24 y 5  9 x 5
12. (5 y 5  5)  5 y
13. (7 x 2  5)  (3x 2  8)
14. (8x 3  4)  (5 x 3  1)
15. (2 x  4)(3x  2)
16. (5 y 5  5)  (5 y 3  2)
17. (7 x 2  4 x  12)  (5x 2  11)
18. (2 x 3  4 x  1)  (3x  2)
6
Algebra 1
Review- Polynomial Operations Quiz
Name:____________________________________
Date:_____________________________________
Objective: To review the material that you will be tested on as part of a quiz. These topics are in the outline
below:
I.
Working with Monomials
a. Addition, subtraction, multiplying and dividing monomials
b. Degree
II.
Operations with Polynomials
a. Degree
b. Adding Polynomials
c. Subtracting Polynomials
d. Multiplying Polynomials
e. Dividing Polynomials
__________________________________________________________________________________________
Below you will find a sample f the types of problems you can expect to see on the test.
I.a.
Perform the four basic operations ( ,,, ) on the following sets of monomials.
1. 6a, 12a
2. 3 xy, 6 xy
3. 25 x 2 yz 2 , 5 x 2 yz
Multiply the following Monomials
4. (3a 2 ) 3 , 4a
I.b.
5. 34 xyz, 10 x 3 yz 5
Determine the value of the coefficients and the degrees of the following monomials.
6. 2a 2
7. 5
8. 4xy 2
7
II.a.
Determine the degree of the following polynomials.
9. 3 x 2  6 x  9
10. 14 xyz 2  x 5
II.b&c. Combine and simplify the following.
12. 3c  7d  (4c  3d )  5c  8d  (d  c)
II.d.
II.e.
11. 1  x
13. Subtract 3 p 2  9 p  7 from  2 p 2  7
Multiply the following polynomials and simplify.
14. 3c(4c  3d )
15. ( x 2  1)(3x  2)
16. ( x  2)(2 x 2  x  4)
Divide the following polynomials and simplify.
17. Divide 3x  6 x  12 by 3
2
6 s 3  8s 2  2 s
18.
2s
19. (30  m  m 2 )  (5  m)
8
9
Algebra 1
Lesson: Factoring Out a GCF
Name:____________________________________
Date:_____________________________________
Objective:
To learn to factor out a Greatest Common Factor.
Do Now: Find the prime factorization of 242
__________________________________________________________________________________________
What is a Factor?
What is a GCF?
How do you factor out a GCF?
__________________________________________________________________________________________
Examples:
1.
5x2  5x
2.
10 xy  15x 2 y 2
3.
c 3  c 2  2c
10
Algebra 1
Lesson- Factoring Perfect Square Trinomials
Name_____________________________________
Date _____________________________________
Objective: To discover patterns within perfect square trinomials.
Factor the following, noticing any patterns that arise.
Use the space at the right to record your observations.
1.
x 2  8 x  16  (
)(
)
2.
x 2  16 x  64  (
)(
)
3.
x 2  12 x  36  (
)(
)
4.
x 2  10 x  25  (
)(
)
Notes:
Use the patterns you recognized in the problems above to factor the more difficult problems below. Define and
then summarize the procedure for factoring perfect square trinomials in the right margin.
5.
16 x 2  8 x  1  (
)(
)
6.
9x 2  6x  1 
(
)(
)
7.
4 x 2  20 x  25  (
)(
)
8.
16 x 2  24 x  9  (
)(
)
9.
9 x 2  30 x  25  (
)(
)
Fill in the blanks so that each expression is a perfect square trinomial, then factor.
10.
x 2  18x  __________  (
)(
)(
)2
11.
x 2  __________  121  (
)(
)(
)2
12.
9 x 2  __________  36  (
)(
)(
)2
13.
________ x 2  20 x  25  (
)(
)(
)2
14.
49 x 2  28x  ________  (
)(
)(
)2
11
Algebra 1
Lesson: Difference of Two Squares
Name:____________________________________
Date:_____________________________________
Objective:
Recognize difference of two squares in binomial form
Factor difference of two squares
DO NOW:
Factor 16 x 2  4 x
__________________________________________________________________________________________
What is a difference of two squares? How do you factor it?
Examples:
Multiply:
( x  2)( x  2)
1.
4.
(1  x)( x  1)
2.
(3  x)(3  x)
3.
(3x  4)(3x  4)
5.
(6 x  13)(6 x  13)
6.
( 3  x)( 3  x)
Factor:
1.
x2  1
2.
4x2  1
5.
16 x 4  9
6.
1 4
x 1
16
3.
9 y 2  16
4.
1 2 16
x 
4
25
12
Algebra 1
Lesson: Factoring Trinomials
Name:____________________________________
Date:_____________________________________
Objective:
Recognize trinomials that are not differences of two squares or perfect square trinomials
Factor any trinomial with a leading coefficient of 1.
DO NOW:
Factor 4 x 2  1
__________________________________________________________________________________________
Polynomials:
There are 3 types of polynomials that we will focus on:
1.
2.
3.
DOTS
Perfect Squares
Everything else
Both a DOTS and a Perfect Square Trinomial can be identified by the fact that the 1st and last terms are always
perfect squares. If there are only 2 terms and a – separates those terms, then it is DOTS. If there are 3 terms
and the 1st and last are perfect squares, then it is a perfect square trinomial.
But what if it is neither…?
To Factor any trinomial, you must know the following cases:
Case 1: (+ +) x 2  5 x  6
Case 2: (– +) n 2  7 n  12
Case 3: (+ –) y 2  5 y  6
Case 4: (– –)
x 2  2 x  35
13
Algebra 1
Lesson: Factoring Completely
Name:____________________________________
Date:_____________________________________
Objective:
To be able to factor any polynomial completely
DO NOW:
Factor x 2  7 x  12
__________________________________________________________________________________________
Process for Factoring Completely: (Usually Multiple Steps)
Examples:
1.
by 2  4b
3.
6x2  6 y 2
2.
3x 2  6 x  24
4.
10a 3  20a 2  10a
14
Algebra 1
WKST- Mixed Factoring
Name_____________________________________
Date _____________________________________
Objective: To evaluate prior factoring skills
Factor completely. If the polynomial can’t be factored, write prime.
1)
x 2  64
2)
x 2  11x  28
3)
x 2  7 x  18
4)
x 2  4 x  12
5)
x 3  225 x
6)
7)
x 2  6x  2
8)
9)
 x 2  3x  2
5 y 5  135 y
12 x 2 y 3  18 x 3 y 4
15
Algebra 1
Review- Factoring
Name_____________________________________
Date _____________________________________
Objective:
To Review Factoring of:
 Perfect square trinomials
 Difference of two squares
 Sum of two squares
 Factoring a Trinomial
Perfect Square Trinomial
To factor a perfect square trinomial follow the steps below:
1) Create two empty binomials as indicated on the right
(
2) Take the square root of the first term of the given trinomial
3) Take the square root of the last term of the given trinomial
4) Take the result of step 2 and put in the 1st position in each binomial
5) Take the result of step 3 and put in the 2nd position in each binomial
6) The signs in the binomials should be the same as the middle term of the binomial
Example 1:
Answer:
)(
)
factor : 4 x 2  12 x  9
(2x - 3)2
Difference of Two Squares a 2  b 2
To factor a difference of two squares follow the steps below:
1) Create two empty binomials as indicated on the right
(
)(
)
st
2) Take the square root of the first term of the given binomial and put in the 1 position of each
binomial
3) Take the square root of the last term of the given binomial and put in the 2nd position of each
binomial
4) Make one binomial a sum and the other binomial a difference
Example 2:
Answer:
9 x 2 y 2  16
(3xy - 4) (3xy + 4)
a 2  b2
Sum of Two Squares
The sum of squares is not factorable
16
Sum of Two Cubes
To factor a sum of two cubes follow the steps below:
1) Create an empty binomial and an empty trinomial as indicated on the right (
)(
2) Take the cube root of the first term of the given expression and put it in the 1st position in the
binomial. Square it and put it in the first position of the trinomial.
**note- ignore all signs until the last step**
3) Take the cube root of the last term of the given expression and
a. put it in the 2nd position in the binomial
b. square it and put it in the last position of the trinomial
4) Find the product of the terms in the binomial and put it in the middle position of the trinomial
5) Arrange the signs as follows ( + )( − + )
)
Factoring a Trinomial
Steps:
1) Check for a GCF. If one can be removed, remove it.
2) Determine the values that multiply to the “last term” that also add to the “middle” term. Signed
numbers should be used here.
Example 6:
x 2  7 x  18
( x  9)( x  2)
17
Algebra 1
Review- Mixed Polynomials and Factoring
Name:____________________________________
Date:_____________________________________
Answer each of the following neatly and completely in the space provided.
1.
Multiply:
3x 2 y  2 x 3 y 2
2.
Multiply:
3x(2 x  6)
3.
Multiply:
(3x  2)( 2 x  1)
4.
Multiply:
x( x  1)( 2 x  2)
5.
Add:
3x 2  2 xy
6.
Add:
3x  (2 x  6)
7.
Add:
(3x  2)  (2 x  1)
8.
Add:
x  ( x  1)  (2 x  2)
9.
Subtract:
3x 2  2 xy
10.
Subtract:
3x  (2 x  6)
11.
Subtract:
(3x  2)  (2 x  1)
12.
Subtract:
x  ( x  1)  (2 x  2)
13.
Factor:
x 2  10 x
14.
Factor:
25 x 2  16
15.
Factor:
a 2  4a  21
16.
Factor:
17.
Multiply:
( x  2)( x  2)
18.
Multiply:
(3  x) 2
19.
Multiply:
(3x  4)(3x  4)
20.
Multiply:
(1  x)( x  1)
21.
Multiply:
(6 x  13) 2
22.
Multiply:
( 4  x)( 4  x)
23.
Factor:
x2  2x  1
24.
Factor:
x2  6x  9
25.
Factor:
9 y 2  24 y  16
26.
Factor:
16 x 4  24 x 2  9
x 2  10 x  25
18
Algebra 1
Lesson- Graphing Quadratic Functions
Name:____________________________________
Date:_____________________________________
Objective:
Do Now:
To graph a quadratic function using the
 Roots
 Axis of symmetry
 Vertex
y
Graph the following function.
f ( x)  3x  1
x
Roots:
Axis of symmetry:
Vertex:
__________________________________________________________________________________________
Graph each of the following on the same set of axes by finding the intercepts, axis of symmetry and vertex.
1. y  x 2  4 x
2. y   x 2  2 x  5
3. y  x 2  4 x  4
Based on the graphs above, is there a shortcut for determining if the parabola opens up or down?
19
20
Algebra 1
Lesson- Zero Product Rule
Name:____________________________________
Date:_____________________________________
Objective:
To apply the zero product method for finding the roots of a quadratic function.
Do Now:
Create a table of values to graph the following function:
y  x 2  14 x  1
Zero product rule:
Find the roots of the given quadratic using the zero product method:
1. x 2  4 x  3  0
2. x 2  7 x  6  0
3. x 2  10 x  0
4. 3 x 2  x
5. x( x  4)  5
6. x 2  2( x  12)
7.
x
3

2 x 1
9. x 2  25  0
8.
3x x 2

2
4
10. 16 x 2  64
21
Algebra 1
HW- Solving for the Roots of Quadratics
Name:____________________________________
Date:_____________________________________
Answer each of the following neatly and completely.
Find the roots for each of the following equations using both an a graphing and algebraic approach.
1.
x 2  81
2.
s2  s  0
3.
y2  6 y
4.
1 2 7
x  x0
2
6
5.
z 2  5z  4  0
6.
x 2  11x  24  0
7.
x 2  8 x  15  0
8.
y2  y  0
9.
x 27

3
x
10.
4 x 2  36  0
11.
x 2  25
12.
x 2  3 x  10  0
22
Algebra 1
Lesson: Quadratic Word Problems
Name:____________________________________
Date:_____________________________________
Objective:
To learn to interpret quadratic word problems, write symbolically and solve.
Do Now: Solve for x:
x
x2

x 1
2
__________________________________________________________________________________________
Method:
__________________________________________________________________________________________
1
You have to make a square-bottomed, unlidded box with a height of three inches and a volume of
approximately 42 cubic inches. You will be taking a piece of cardboard, cutting three-inch squares from
each corner, scoring between the corners, and folding up the edges. What should be the dimensions of
the cardboard, to the nearest quarter inch?
2.
The product of two consecutive negative integers is 1122. What are the numbers?
3.
You have a 500-foot roll of fencing and a large field. You want to construct a rectangular playground
area. What are the dimensions of the largest such yard? What is the largest area?
23
4.
In right triangle ABC, AB is two more than BC, AC is one more than BC. Find the length of each side
of right triangle ABC.
5.
The product of two consecutive ODD integers is 99. Find the integers.
6.
The length of a rectangular garden is 4 meters more than the width. The area of the garden is 60 square
meters. Find the dimensions of the garden.
7.
a.
b.
c.
If the length of a side of a square is represented by x + 6, express its area in terms of x.
If the dimensions of a rectangle are represented by 3x + 2 & x – 2, express its area in terms of x.
If the area of the square is equal to the area of the rectangle, find x and the value of the
dimensions of each shape.
24
Algebra 1
Name:________________________________________
Wkst- Quadratic Word Problems
Date:_____________________________________
A rectangular dog pen is to be made along an
existing fence. The total length of the new fence
is to be 100 feet and the width is x.
Existing fence
x
pen
x
a) Express, in simplest from, the area of the pen in terms of x.
b) What is the width that gives the maximum area? Show or explain how you arrived at your answer.
c) What is the maximum area?
25
Another version of the fence problem… The owner takes
down the fence and now wants to use the 100 feet of fencing
to make two adjacent dog pens against the existing wall. See
the diagram at the right.
Existing wall
x
x
x
a) If x is the length of fence perpendicular to the existing wall, express, as a function of x, the length of the fence parallel
to the wall.
b) Express, as a function of x, the total area of both pens.
c) What is the value of x (to the nearest tenth) that gives the maximum area? Show or explain how you arrived at your
answer.
d) What is the maximum total area(to the nearest whole unit)?
26
Algebra 1
Lesson- Solving quadratic linear systems
Name:____________________________________
Date:_____________________________________
Objective:
To learn how to solve systems of equations involving a line and a parabola.
Do Now:
Solve for x:
2 x 2  3x  9  0
__________________________________________________________________________________________
Solving a quadratic linear system
Graphical Process:
Algebraic Process:
Examples:
y  x 2  2x  3
1.
2x  y  2
2.
y  x 2  2x  1
y  2x  3
3.
y  x2 1
y  x 1
27
Algebra 1
Review- Quadratics Test
Name:____________________________________
Date:_____________________________________
Objective:
To review the following topics for a test on quadratics.
I.
Solving for the roots of quadratics
a.
by factoring
b.
by graphing calculator
II.
Graphing quadratics
a.
axis of symmetry
b.
vertex
c.
roots
d.
maximum/minimum
III.
Quadratic/Linear Systems
a.
Graphic Solution (including calculator)
b.
algebraic solution
IV.
Quadratic word problems
Answer each of the following on separate paper.
x 2  81  0
1.
Solve by factoring:
y 2  3 y  10  0
2.
Solve by factoring:
3.
Solve by factoring:
14 x  x 2  49
4.
Solve by GC:  3x 2  5 x  4  0
5.
Write the equation of the axis of symmetry, tell whether the turning point is a maximum or minimum,
find the coordinates of the vertex, find the roots (if they exist, and graph the parabola for: y  x 2  10 x
6.
Write the equation of the axis of symmetry, tell whether the turning point is a maximum or minimum,
find the coordinates of the vertex, find the roots (if they exist, and graph the parabola for:
1
1
y  x 2  3x 
2
2
7.
Write the equation of the axis of symmetry, tell whether the turning point is a maximum or minimum,
find the coordinates of the vertex, find the roots (if they exist, and graph the parabola
1
for: y  x 2 (4  x  4)
2
y  x2
Find the solution of the following quadratic/linear system algebraically and graphically:
y  x2
8.
y  x2  3
9.
Find the solution of the following quadratic/linear system algebraically and graphically:
10.
Find the solution of the following quadratic/linear system algebraically and graphically:
11.
In a rectangle, the measure of the height is 5 less than the measure of the base, and the area is 14. Write
and solve an equation to find the dimensions of the rectangle.
y  x  2
y  x2 1
y  x3
28
12.
The perimeter of a right triangle is 30 inches and the length of its hypotenuse is 13 inches.
a.
If x represents the length of one leg, represent the length of the other leg in terms of x.
b.
Write and solve an equation to find the lengths of the legs of the right triangle.
13.
The perimeter of a rectangle is 27 cm and its area is 35 square cm. Find the length and width of the
rectangle.
y
y
x
y
x
y
x
x
29