• Study Resource
• Explore

Survey

* Your assessment is very important for improving the work of artificial intelligence, which forms the content of this project

Document related concepts

Elementary mathematics wikipedia, lookup

Fundamental theorem of algebra wikipedia, lookup

Factorization of polynomials over finite fields wikipedia, lookup

Line (geometry) wikipedia, lookup

System of polynomial equations wikipedia, lookup

Vincent's theorem wikipedia, lookup

Recurrence relation wikipedia, lookup

Algebra wikipedia, lookup

Polynomial wikipedia, lookup

Horner's method wikipedia, lookup

Transcript
```Algebra 1 Key Vocabulary Words
Chapter 9
Section 9.1: monomial
degree of monomial
polynomial
degree of polynomial
binomial
trinomial
Section 9.2: algebra unit-tile (not a Key Vocabulary)
algebra x-tile (not a Key Vocabulary)
algebra x2-tile (not a Key Vocabulary)
distributive property of multiplication (see Ch 2.5)
FOIL pattern
area of rectangle (not a Key Vocabulary)
square of a binomial
difference of two squares (sum and difference pattern)
Section 9.3: perfect square trinomial
difference of two squares
Section 9.4: roots
vertical motion
zero-product property
Section 9.5: zero of a function
Section 9.6: Trinomial (see Ch 9.1)
Section 9.7: perfect square trinomial (see Ch 9.3)
Section 9.8: factor completely
common factor
product of binomial factors
Permission for Use Granted by Dr. Kate Kinsella 10/08
Word
Monomial
9.1
Meaning
DEF:
A monomial is a
number, a variable, or the
product of a number and
one or more variables
with whole number
exponents.
Examples
Oral Practice
The following are monomials:
x
m2
The following are examples of polynomials: y3,
5mn, 4 and p.
____ and ____ are also examples of monomials.
-2xy
5
-2b2c
Writing Practice:
The following are examples of monomials: 3c, -4f2, 5xy, and 6.
The following are examples of monomials: ________________.
Degree of
Monomial
9.1
The Degree of a
Monomial is the sum of
the exponents of the
variables in the
monomial.
Monomial
Degree
x
1
m3
3
2xy
2
-3v12u2
14
5
0
The degree of the monomial 2x3 is 3 because the
exponent of the variable x is 3.
The degree of the monomial 2xy4 is 5 because the
exponent of the variable x is 1 and the exponent
of the variable y is 4 and 4 + 1 = 5
The degree of the monomial 12p3k4 is 7 because
the exponent of the variable p is 3 and the
exponent of the variable k is 4 and 3 + 4 = 7
Writing Practice:
The degree of the monomial -x7 is 7 because the exponent of the variable x is 7.
The degree of the monomial ______ is _____ because _______________________.
Permission for Use Granted by Dr. Kate Kinsella 10/08
Word
Polynomial
9.1
Meaning
A polynomial is a
monomial or a sum of
monomials.
Examples
Oral Practice
Below are examples of polynomials
2x3 – 3 is a polynomial.
2x (monomial)
-x + 3y – 5 is a polynomial
3x + y (sum of 2 monomials)
_______________ is a polynomial,
5m2 – 3m + 2 (sum of 3 monomials)
Writing Practice:
3f – 5 is a polynomial.
_______ is a polynomial.
Degree of
Polynomial
9.1
The Degree of a
Polynomial is the
greatest degree of its
terms.
Polynomial
Degree
x+5
1
m3 – 2m2 + 3
3
2xy + 5z5 – 1
5
-3v12 + 6u4v10 + w12
14
2x2 + 3xyz
3
The polynomial 3x4 – 5x3 + 2x3 – 1 is a 4th degree
polynomial.
The polynomial 2xy – y + 1 is a 2nd degree
polynomial.
The polynomial ______________ is a ___ degree
polynomial.
Writing Practice:
The polynomial 2x5 – 3x2 – 1 is a 5th degree polynomial.
The polynomial __________ is a ___ degree polynomial.
Permission for Use Granted by Dr. Kate Kinsella 10/08
Word
9.1
Meaning
DEF:
When a polynomial is
written so that the
exponents of a variable
decrease from left to
right, the coefficient of
the first term is the
Examples
Oral Practice
The leading coefficient of -2x5 + 3x3 –
2x + 5 is –2.
The leading coefficient of 4v3 – 2v2 + v – 7 is 4.
The leading coefficient of 5m4 – 3m2 is
5.
The leading coefficient of 2x3 – 5x + 1 is ___.
Writing Practice:
The leading coefficient of 8m2 – 5m + 1 is 8.
The leading coefficient of ______________ is ____.
Binomial
9.1
DEF:
A polynomial with two
terms is called a
binomial.
The following are examples of
binomials.
Since the polynomial 2x + 8 has two terms then
it is a binomial.
2x + 3
-5mn + 6n
Since the polynomial 3m + 5n has _______ then
it is a _________.
x+y
-2x3 + 6xy
Writing Practice:
Since the polynomial –5v + 7 has two terms then it is a binomial.
Since the polynomial _______ has two terms then it is a ________.
Permission for Use Granted by Dr. Kate Kinsella 10/08
Word
Trinomial
9.1
Meaning
A polynomial with three
terms is called a
trinomial.
Examples
The following are examples of
trinomials
Oral Practice
Since the polynomial 2x – y + 3 has three terms
then it is a trinomial.
2x + 3y – 1
-5mn + 6n + 7
Since the polynomial ________ has ____ terms
then it is a _______.
x+y+z
-2x3 + 6xy – 7
Writing Practice:
Since the polynomial 5m + 6n – 1 has three terms then it is a trinomial.
Since the polynomial ________ has ____ terms then it is a _______.
FOIL Pattern
9.2
This pattern which means
F – first, O – outer, I –
inner, and L – last
helps you remember how
to use the distributive
property to multiply
binomials.
To get the product we use FOIL pattern
described as:
F
L
For the factors (x – 6)(x + 7), identify the FOIL
Products
First term products: (x)(x)
Outer term products: (x)(7)
Inner term products: (-6)(x)
Last term products: (-6)(7)
(2x + 3) (x + 4)
I
O
First term products: (2x)(x)
Outer term products: (2x)(4)
Inner term products: (3)(x)
Last term products: (3)(4)
For the factors (3x + 5)(x – 2), identify the FOIL
Products
First term products: (3x)(x)
Outer term products: (3x)(-2)
Inner term products: (5)(x)
Last term products: (5)(-2)
Writing Practice:
For the factors (x + 1)(x + 8), identify the FOIL Products. F – (x)(x) O – (x)(8) I - (1)(x) L – (1)(8)
For the factors (_______)(_______) identify the FOIL Products
Permission for Use Granted by Dr. Kate Kinsella 10/08
Word
Square of a
Binomial
9.2
Meaning
DEF:
Square of a binomial
Pattern:
(a + b)2 = a2 + 2ab + b2
(a – b)2 = a2 – 2ab + b2
Examples
To use the pattern, identify the 1st term
a and the 2nd term b. Plug into the
formula and simplify:
2
(2x + 3)
a = 2x and b = 3
(2x)2+2(2x)(3) + (3)2= 4x2 + 12x + 9
PRODUCT
Oral Practice
Use the pattern to get the product
(x – 5)2
a = x and b = -5
(x)2+2(x)(-5) + (-5)2= x2 – 10x + 25
PRODUCT
Use the pattern to get the product
(2x – 7)2
a = 2x and b = -7
( )2+2( )( ) + ( )2= ____________
PRODUCT
Writing Practice:
Find the product: (3x + 5)2 = 9x2 + 30x + 25
Find the product: (________)2 = _____________
Difference of 2
Squares
9.2
DEF:
(a + b)(a – b) = a2 – b2
To use the pattern, identify the 1st term
a and the 2nd term b and make sure
that they assume the (a + b)(a – b)
form.
(2x + 3)(2x – 3) = ?
a = 2x and b = 3
To use the pattern, identify the 1st term a and the
2nd term b and make sure that they assume the
(a + b)(a – b) form.
(x – 3)(x + 3) = ?
a = x and b = 3
Plug into a and b and simplify:
Plug into a and b and simplify:
2
2
2
(2x) – (3) = 4x – 9 (product)
(x)2 – (3)2 = x2 – 9 (product)
Find the product using the pattern:
(2x + 7)(2x – 7) = ( )2 – (
)2 = ________
Writing Practice:
Find the product using the pattern:
(x + 9)(x – 9) = (x)2 – (9)2 = x2 – 81
(_______)(_______) = ( )2 – (
)2 = ________
Permission for Use Granted by Dr. Kate Kinsella 10/08
Word
Perfect Square
Trinomial (PST)
9.3
Meaning
Examples
DEF:
a2 + 2ab + b2 = (a + b)2
2
2
a – 2ab + b = (a – b)
2
The following is a perfect square
trinomial:
4x2 + 4x + 1 because the middle term
is a product of the square root of the
first and last term and 2.
Square root of 4x2 is 2x and square
root of 1 is 1
Oral Practice
The trinomial x2 – 6x + 9 is a PST because the
middle term is 2(x)(3) and therefore can be
factored as (x – 3)2
The trinomial __________ is a PST because the
middle term is _____ and therefore can be
factored as _________.
2(2x)(1) = 4x, therefore can be
factored as
(2x + 1)2
Writing Practice:
The trinomial 9x2 – 12x + 4 is a PST because the middle term is 2(3x)(2) and therefore can be factored as (3x – 2)2
The trinomial ____________is a PST because the middle term is _______ and therefore can be factored as ________.
Roots
9.4
DEF:
The solutions of
equations. (Such as an
equation where one side
is zero and the other side
is a product of polynomial
factors)
(x + 3)(x – 2) = 0
(x + 5)(x + 7) = 0
Since x = -5 or x = -7
Since x = -3 or x = 2
The roots of the equation are –3
and 2
The roots of the equation are –5 and –7
(x – 8)(x + 5) = 0
Since x = 8 or x = -5
The roots of the equation are _________.
Writing Practice:
(x)(x + 2) = 0, Since x = 0 or x = -2. The roots of the equation are 0 and –2.
______________ = 0, Since ______ or ______. The roots of the equation are __________.
Permission for Use Granted by Dr. Kate Kinsella 10/08
Word
Solutions of
9.4
Meaning
DEF:
The solutions of a
values of the x that
satisfies the equation. It
varies from none, one or
two solutions.
Examples
Equations and their Solutions:
x2 – 1 = 0
The solutions of this quadratic equation
are x = 1. When these values are
plugged into the equation, it makes a
true statement.
Oral Practice
x2 – 2x + 1 = 0
The solution of this quadratic equation is x = 1.
When 1 is plugged in the equation it makes the
equation true.
x2 – 4x + 3 = 0
The solution of this quadratic equation is _____
When ____ is plugged in the equation it makes
the equation true.
Writing Practice:
The solution to x2 – 9 = 0 is 3. When these values are plugged in the equation, it makes the equation true.
The solution to ________ is ___. When these values are plugged in the equation, it makes the equation true.
The Vertical Motion Model
Vertical Motion
DEF:
An equation for an object thrown upwards with an
Model
The height h (in feet) of a projectile can initial velocity of 11 ft/sec is given as:
9.4
The Vertical Motion
be modeled by:
Let v = 10 ft/sec and s = 0
Model can describe the
path of a projectile. A
h = -16t2 + vt + s
h = -16t2 + (11)t + (0)
projectile is an object
h = -16t2 + 11t
thrown into the air but
Where t is time in seconds the object
has no power to keep
has been in the air, v is the initial
itself in the air.
velocity, and s is the initial height in
An equation for an object thrown upwards with an
feet.
initial velocity of ________ is given as:
Let v = _______ and s = 0
The model can be used to find an
equation that gives the height of an
h = -16t2 + (___)t + (__)
object as a function of time (in
h = ______________
seconds) given the initial velocity.
(s = 0 at this point)
Writing Practice:
An equation for an object thrown upwards with an initial velocity of 12 ft/sec is given as: h = -16t2 + 12t
An equation for an object thrown upwards with an initial velocity of _______ is given as: _____________
Permission for Use Granted by Dr. Kate Kinsella 10/08
Word
Meaning
Zero-Product
Property
9.4
DEF:
Let a and b be real
numbers. If ab = 0, then
a = 0 or b = 0.
Examples
The zero product property states that:
If (x +3)(x – 2) = 0,
Oral Practice
Fill in the correct conclusion:
If (2x – 5)(x + 8) = 0,
Then (2x – 5)= 0 or (x + 8) = 0
Then (x +3) = 0 or (x – 2) = 0
Fill in the correct conclusion:
If (x – 9)(3x – 7) = 0,
Then ______________________
Writing Practice:
If (x + 5)(x + 7) = 0, then (x + 5) = 0 or (x + 7) = 0
If (
)(
) = 0, then _____________________
Zero of a Function
9.5
DEF:
The zero of a function
y = f(x) is a value of x for
which f(x) = 0. (or y = 0)
Given:
Given:
F(x) = x – 5
F(x) = 2x – 1
Setting F(x) = 0 and solving for x, the value of
x=5
Therefore, 5 is a zero of the function because
F(5) = 0
Setting F(x) = 0 and solving for x, the
value of x = 1/2
Therefore, 1/2 is a zero of the
function because F(1/2) = 0
F(x) = 2x + 6
Setting F(x) = 0 and solving for x, the value of
x = -3
Therefore, ____________________ because
________
Writing Practice:
If F(-2) = 0, then –2 is a zero of the function.
If _______, then ___ is a zero of the function.
Permission for Use Granted by Dr. Kate Kinsella 10/08
Word
Factor Completely
9.8
Meaning
DEF:
A factorable polynomial
with integer coefficients is
said to be factored
completely if it is
written as a product of
unfactorable polynomials
with integer coefficients.
Examples
Given 2x2 – 5x – 12 = (2x + 3)(x – 4)
Since it is written as (2x + 3)(x – 4)
with integer coefficients, then the
polynomial 2x2 – 5x – 12 is factored
completely.
Oral Practice
Given 15x2 + 11x + 2 = (3x + 1)(5x + 2)
Since it is written as (3x + 1)(5x + 2) with
integer coefficients, then the polynomial
15x2 + 11x + 2 is factored completely.
Given 5x2 – 14x – 3 = (x – 3)(5x + 1)
Since it is written as (x – 3)(5x + 1) with
integer coefficients, then the polynomial
5x2 – 14x – 3 is factored completely.
Writing Practice:
Given 2x2 + 5x – 3 = (x + 3)(2x – 1). Since the factors have integer coefficients, then it is factored completely.
Given __________ = ____________. Since the factors have _____________, then ___________________.
Common Factor
9.8
DEF:
A common factor is a
whole number that is a
factor of each number.
Given the numbers and their factors:
25: 1, 5, 25
10: 1, 2, 5, 10
The number 5 is a common factor.
Given the numbers and their factors:
12: 1, 2, 3, 4, 6, 12
20: 1, 2, 4, 5, 10, 20
The numbers 3 and 4 are common factors.
Given the numbers and their factors:
24: 1, 2, 3, 4, 6, 8, 12, 24
36: 1, 2, 3, 4, 6, 9, 12, 36
The numbers _____________ are common
factors.
Writing Practice:
The common factor(s)of 12 and 36 are 2, 3, 4, 6, and 12
The common factor(s)of ____ and ____ are _____________
Permission for Use Granted by Dr. Kate Kinsella 10/08
Word
Product of
Binomial Factors
9.8
Meaning
DEF:
If a polynomial is factored
completely and the
factors are written as a
binomial, the factors are
said to be written as
product of binomial
factors.
Examples
The factors of 15x2 + 11x + 2 are
written as product of binomial
factors as (3x + 1)(5x + 2) because
the factors are binomials.
Oral Practice
The factors of 5x2 – 14x – 3 are written as
product of binomial factors as (x – 3)(5x + 1)
because the factors are binomials.
The factors of ___________are written as
product of binomial factors as ____________
because the factors are binomials.
Writing Practice:
The factors of 2x2 + 5x – 3 are written as product of binomial factors as (x + 3)(2x – 1) because the factors are binomials.
The factors of __________ are written as product of binomial factors as ____________ because the factors are binomials.
Permission for Use Granted by Dr. Kate Kinsella 10/08
```
Related documents