Transcript Addition Property (of Equality)

MATH!!! EXAM PREP!!!!
ConoR
RoweN
Addition
Property (of Equality)
.
If the same number is added to both sides of an equation,
the two sides remain equal.
if (x) = (y), then (x) + z = (y) + z.
Multiplication Property (of Equality)
For all real numbers a and b , and for c ≠ 0 , a = b
equivalent to
ac = bc
is
Reflexive Property (of Equality)
D=D
Symmetric Property (of Equality)
if h = b, then b = h
Transitive Property (of Equality)
if a = b and b = c, then a = c .
Associative Property of Addition
(a + b) + c = a + (b + c)
Associative Property of Multiplication
When three or more numbers are multiplied, the
product is the same regardless of the grouping of
the factors. For example (2 * 3) * 4 = 2 * (3 * 4)
Commutative Property of Addition
When 2 numbers are added together, the sum is
the same regardless of the order of the addends.
example 4 + 2 = 2 + 4
Commutative Property of Multiplication
When two numbers are multiplied together, the
product is the same regardless of the order of
the multiplicands. For example 4 * 2 = 2 * 4
Distributive Property (of Multiplication over
Addition)
multiplying a sum by a number gives the same
result as multiplying each addend by the number
and then adding the products together.
4 × (2 + 3) = 4 × 2 + 4 × 3
Prop of Opposites or Inverse Property of Addition
EXAMPLE:: (8 - 8 = 0)
Inverse Property of Addition---- Any number added
to its opposite integer will always equal ZERO. If
using addition the order of the numbers doesn’t
matter
[Ex. 3 + (-3) = 0 or (-3) + 3 = 0]
Prop of Reciprocals or Inverse Prop. of Multiplication
For every number, x, except zero, has a multiplicative
inverse,1/x. EXAMPLE x * (1/x) = 1
Identity Property of Addition
the sum of zero and any number or variable is the number
or variable itself.
EXAMPLES:::: 4459907 + 0 = 4459907, - 1178 + 0 = - 1178,
y + 0 = y are few examples illustrating the identity property
of addition.
Identity Property of Multiplication
the product of 1 and any number or variable is the number or variable
itself.
EXAMPLES::::: 3 × 1 = 3, - 118997006 × 1 = - 118997006, ybbh × 1 = y bbh
few examples illustrating the identity property of multiplication.
Multiplicative Property of Zero
The product of 0 and any number results in 0.That is, for
any real number a, a × 0 = 0.
Closure Property of Addition
the sum of any two real numbers equals another real number.
2, 5 = real numbers.
2 + 5 = 7(real number).
Closure Property of Multiplication
the product of any two real numbers equals another real number.
4, 7 = real numbers.
4 × 7 = 28(real number)
Product of Powers Property
to multiply powers having the same base, add the exponents.
That is, for a real number non-zero a and two
integers m and n, am × an =am+n.
Power of a Product Property
find the power of each factor and then multiply
Power of a Power Property
the power of a power can be found by multiplying the exponents.
for a non-zero real number a and two integers m and n,
(am)n =amn.
Quotient of Powers Property
divide powers that have the same base, subtract the exponents.
That is, for a non-zero real number a and two integers m and n, .
Power of a Quotient Property
the power of a quotient can be obtained by finding the
powers of numerator and denominator and dividing
them.
That is, for any two non-zero real numbers a and b and
any integer m, .
Zero Power Property
any variable with a zero exponent is equal to one
°
example:: c = 1
Negative Power Property
Any variable with a negative exponent is
equal to 1 over its reciprocal.
ExAmPLE:: cˉⁿ = 1/cn
Zero Product Property
the product of two real numbers is zero, then at least one of the
numbers in the product (factors) must be zero.
Cbr = 0 THEN c = 0, b = 0, or r = 0
Product of Roots Property
for any nonnegative real numbers a and b.
EXAmple:::
√ab = √a • √b
Quotient of Roots Property
For any nonnegative real number a
and any positive real number b:
eXaMpLe √a/b = √a/√b
Root of a Power Property
I could not find this property anywhere.
Power of a Root Property
the square root of anything to the
2nd power is that number.
ExAmPLE!!!!!!!!!!!! square root of 121
is 11 and -11.
Now you will take a quiz!
Look at the sample problem and give the name of the
property illustrated.
1. a + b = b + a
Answer:
Commutative Property (of Addition)
Now you will take a quiz!
Look at the sample problem and give the name of the property
illustrated.
2.
4+2=2+4
Answer:
Community Property of Addition
1St power inequalities
Single Sign Inequality Problems
X is less than or equal to 2
Written like: x≤2
Answer this!
How is k less than or equal to 6 written?
k≤6
Conjunction
 Two sentences combined by the word and .
 EXAMPLE: x
›
2 and x
‹
1
Disjunction
Two sentences combined by the word or.
Example:: -3 ‹ x OR x ‹ 4
In the next slides you will review:
Linear
equations in
two
variables
SLOPES
Positive Slope= line running upward (rising)
from left to right
 Negative Slope= lines that are falling from
left to right.
 Horizontal Slope= lines that run left to right
(flat) with no slope (0=slope).
 Vertical Slope= Lines that rise up and down
(undefined= slope)

SLOPE FORMULAS!





General= Ax + By + C= 0
Standard= Ax + By = C
Slope= rise/run
Slope Formula= Y - Y / X - X
Slope-Intercept Form= y=mx + b
(m=slope) (b= y-int.)
 Point-Slope= y – y = m ( x – x )
2
1
1
2
1
1
Example: Graph the line y = 2x – 4.
1. The equation y = 2x – 4 is in the slope-intercept form. So,
m = 2 and b = - 4.
y
2. Plot the y-intercept, (0, - 4).
x
3. slope= 2.
m=
change in y
2
=
1
change in x
4. Start at the point (0, 4).
move 1 space to the right and 2 up
to place the second point on the line.
(1, -2) is also on the line.
(1, -2)
2
(0, - 4)
5. Draw the line through these points that you just
made!
1
y – y1 = m(x – x1) is in point-slope form.
slope (m) passes through the point (x1, y1).
Example:
The graph of the equation
y – 3 = - 1 (x – 4) is a line
2
of slope m = - 1 passing
2
through the point (4, 3).
y
8
4
m=-
1
2
(4, 3)
x
4
8
Linear
SYSTEMS!
Substitution Method
Steps
 Solve one equation for one of the variables
 Substitute this expression in the other equation and
solve for variable.
 Substitute this value in the equation in step 1 and solve
 Check values in both equations
Addition/subtraction
method
 Add or subtract the equations to eliminate one variable
 Solve the resulting equation for the other variable
 Substitute in either original equation to find the value of the first
variable
 Check in both original equations
Dependent, inconsistent,
consistent
FACTORING
The simplest method of factoring a polynomial is to
factor out the greatest common factor (GCF) of each
term.
Example: Factor 18x3 + 60x.
Factor each term.
18x3 = 2 · 3 · 3 · x · x ·
= (2 · 3 · x) · 3 · x · x
x
60x = 2 · 2 · 3 · 5 · x
= (2 · 3 · x) · 2 ·
Find the GCF.
GCF 5= 6x
Apply the distributive
18x3 + 60x = 6x(3x2) + 6x
law to factor the
2
(10)
= 6x(3x +
polynomial.
10)
Check the answer by multiplication.
6x(3x2 + 10) = 6x(3x2) + 6x(10) = 18x3 +
60x
Example:
Factor 4x2 – 12x + 20.
= 4(x2 – 3x +
5)
Check the answer. 4(x2 – 3x + 5) = 4x2 – 12x + 20
GCF = 4.
A common binomial factor can be factored out of
certain expressions.
Example: Factor the expression 5(x + 1) – y(x + 1).
5(x + 1) – y(x + 1) = (x + 1) (5 – y)
Check. (x + 1) (5 – y) = 5(x + 1) – y(x + 1)
difference of two squares
Example: Factor x2 – 9y2.
x2 –
9y2
= (x)2 – (3y)2
Write terms as perfect
squares.
= (x + 3y)(x –
.
3y)
The same method can be used to factor any expression
which can be written as a difference of squares.
Example: Factor (x + 1)2 – 25y 4.
(x + 1)2 – 25y 4 = (x + 1)2 – (5y2)2
= [(x + 1) + (5y2)][(x + 1) – (5y2)]
= (x + 1 + 5y2)(x + 1 – 5y2)
Some polynomials can be factored by grouping terms to
produce a common binomial factor.
Examples: 1. Factor 2xy + 3y – 4x –
6. + 3y – 4x – = (2xy + 3y) – (4x + Group terms.
2xy
6
6)
= y (2x + 3) – 2(2x + 3) Factor each pair of
terms.
= (2x + 3) ( y – 2)
Factor out the common
binomial.
2. Factor 2a2 + 3bc – 2ab – 3ac.
2a2 + 3bc – 2ab –
3ac
= 2a2 – 2ab + 3bc – 3ac Rearrange terms.
= (2a2 – 2ab) + (3bc – 3ac)
Group
terms.
= 2a(a – b) + 3c(b –
Factor.
a)
= 2a(a – b) – 3c(a – b)
b – a = – (a – b).
= (a – b) (2a – 3c)
Factor.
To factor a simple trinomial of the form x2 + bx + c,
express the trinomial as the product of two binomials. For
example,
x2 + 10x + 24 = (x + 4)(x + 6).
Factoring these trinomials is based on reversing the FOIL
process.
Example: Factor x2 + 3x + Express the trinomial as a product of
two binomials with leading term x
2
2.
x + 3x + = (x + a)(x +
and unknown constant terms a and
2
b) F O I
L b.
= +
+ ax + ba Apply FOIL to multiply the binomials.
x=2x2 bx
+ (b + a)x + ba Since ab = 2 and a + b = 3, it
=
x2
+ (1 + 2)x + 1 · 2
follows that a = 1 and b = 2.
Therefore, x2 + 3x + 2 = (x + 1)(x + 2).
Example: Factor x2 – 8x + 15.
x2 – 8x + 15= (x + a)(x + b)
= x2 + (a + b)x + ab
Therefore a + b = -8 and ab = 15.
It follows that both a and b are negative.
Negative Factors of
15
- 1, - 15
Sum
-15
-3, - 5
-8
x2 – 8x + 15 = (x – 3)(x – 5).
Check: (x – 3)(x – 5) = x2 – 5x – 3x + 15
= x2 – 8x + 15.
Factor x2 13
+ 13x36
+
x2 + 13x + = (x + a)(x + b)
36
= x2 + (a + b)x + ab
Therefore a and b are:
Example:
36.
two positive factors of 36
whose sum is 13.
Positive Factors of 36 Sum
1, 36
37
2, 18
3, 12
20
15
4, 9
6, 6
13
12
x2 + 13x + 36= (x + 4)(x + 9)
Check: (x + 4)(x + = x2 + 9x + 4x + 36= x2 + 13x + 36.
9)
A polynomial is factored completely when it is written as a
product of factors that can not be factored further.
Example: Factor 4x3 – 40x2 +
100x.
4x3 – 40x2 +
100x
= 4x(x2 – 10x + 25)
The GCF is 4x.
Use distributive
property to factor out
the GCF.
Factor
the trinomial.
= 4x(x – 5)(x – 5)
Check 4x(x – 5)(x – 5)= 4x(x2 – 5x – 5x + 25)
:
= 4x(x2 – 10x + 25)
= 4x3 – 40x2 + 100x
Factoring complex trinomials of the form ax2 + bx + c, (a  1) can be
done by decomposition or cross-check method.
Example: Factor 3x2 + 8x + 4.
3  4 = 12
Decomposition Method
1. Find the product of
first and last terms
3. Rewrite the middle
term decomposed
into the two
numbers
4. Factor by
grouping in pairs
2. We need to find factors of
12 whose
is 8
sum
3x2 + 2x + 6x + 4
= (3x2 + 2x) + (6x + 4)
= x(3x + 2) + 2(3x + 2)
= (3x + 2) (x + 2)
3x2 + 8x + 4 = (3x + 2) (x + 2)
1, 12
2, 6
3, 4
Example: Factor 4x2 + 8x – 5.
4  5 = 20
We need to find factors of 20
whose difference is 8
Rewrite the middle
term decomposed into
the two numbers
Factor by grouping
in pairs
4x2 – 2x + 10x – 5
= (4x2 – 2x) + (10x – 5)
= 2x(2x – 1) + 5(2x – 1)
= (2x – 1) (2x + 5)
4x2 + 8x – 5 = (2x –1)(2x – 5)
1, 20
2, 10
4, 5
QUADRATIC Equations
QUADRATICS
 A binomial expression has just two terms (usually an x
term and a constant). There is no equal sign. Its
general form is ax + b, where a and b are real numbers
and a ≠ 0.
 One way to multiply two binomials is to use the FOIL
method. FOIL stands for the pairs of terms that are
multiplied: First, Outside, Inside, Last.
 This method works best when the two binomials are in
standard form (by descending exponent, ending with
the constant term).
 The resulting expression usually has four terms before
it is simplified. Quite often, the two middle (from the
Outside and Inside) terms can be combined.
For example:
QUADRATICS
 The opposite of multiplying two binomials is to
factor or break down a polynomial (many termed)
expression.
 Several methods for factoring are given in the text. Be
persistent in factoring! It is normal to try several
pairs of factors, looking for the right ones.
 The more you work with factoring, the easier it will be
to find the correct factors.
 Also, if you check your work by using the FOIL method,
it is virtually impossible to get a factoring problem
wrong.
 Remember! When factoring, always take out any
factor that is common to all the terms first.
 A quadratic equation involves a single
variable with exponents no higher than 2.
 Its general form is
where a,
b, and c are real numbers and
.
 For a quadratic equation it is possible
to have two unique solutions, two
repeated solutions (the same number
twice), or no real solutions.
 The solutions may be rational or irrational
numbers.
 To solve a quadratic equation, ONLY
IF ITS factorable:
 1. Make sure the equation is in the
general form.
 2. Factor the equation.
 3. Set each factor to zero.
 4. Solve each simple linear equation.
To solve a quadratic equation if you can’t factor
the equation:
 Make sure the equation is in the
general form.
 Identify a, b, and c.
 Substitute a, b, and c into the quadratic
formula:
 Simplify.
Tips & Tricks
 The cool, easy thing about the
quadratic formula is that it works on
any quadratic equation when put in
the form general form.
 When having trouble factoring a problem,
the quadratic formula might be quicker.
 be sure and check your solution in the
original quadratic equation.
Simplifying Rational
Expressions
 The objective is to be able to simplify a
rational expression
5
x2
3x
x 9
2
 ALWAYS, ALWAYS,
ALWAYS Divide out
the common factors
 Factor the
numerator and
denominator and
then divide the
common factors
Dividing Out Common
Factors
Step 1 – Identify any factors which are common to both the
numerator and the denominator.
5x
5( x  7)
The numerator and denominator
have a common factor.
The common factor is the five.
Dividing Out Common
Factors
Step 2 – Divide out the common factors.
The fives can be divided since 5/5 = 1
The x remains in the numerator.
The (x-7) remains in the denominator
5x
5( x  7)

x
x7
Factoring the
Numerator
and Denominator
Factor the numerator.
Factor the denominator.
Divide out the common factors.
Write in simplified form.
3x  9 x
2
12 x
3
Factoring
Step 1: Look for common factors to
both terms in the numerator.
3x  9 x
2
12 x
3 is a factor of both 3 and 9.
X is a factor of both x2 and x.
3
Step 2: Factor the numerator.
3x  9 x
2
12 x
3
3x ( x  3)
12 x
3
Factoring
Step 3: Look for common factors to the
terms in the denominator and factor.
3x  9 x
2
12 x
3
The 12 can be factored into 3 x 4.
The x3 can be written as x • x2.
3x  9 x
2
12 x
The denominator only has one term.
The 12 and x3 can be factored.
3
3x(x  3)
2
3 4  x  x
Divide and
Simplify
Step 4: Divide out the common factors. In this
case, the common factors divide to become 1.
3x ( x  3)
3 4  x  x
2
Step 5: Write in simplified form.
x3
4x
2