MATH!!! EXAM PREP!!!! ConoR RoweN. Addition Property (of Equality) Multiplication Property (of Equality). If the same number is added to both sides of.

MATH!!! EXAM PREP!!!!

ConoR RoweN

Addition Property (of Equality)Addition Property (of Equality)

Multiplication Property (of Equality)Multiplication Property (of Equality)

. .

If the same number is added to both sides of an If the same number is added to both sides of an equation, the two sides remain equal. equation, the two sides remain equal. if if (x)(x) = = (y) (y), then (, then (x)x) + + z = (y) + z. z = (y) + z.

For all real numbers  For all real numbers  aa  and    and  bb , and for   , and for  cc ≠ 0 , ≠ 0 , aa = = bb     is     is

equivalent to     equivalent to     acac = = bcbc  

Reflexive Property (of Equality)Reflexive Property (of Equality)

Symmetric Property (of Equality)Symmetric Property (of Equality)

Transitive Property (of Equality)Transitive Property (of Equality)

D = D D = D

if h = b, then b = hif h = b, then b = h

if a = b and b = c, then a = c .if a = b and b = c, then a = c .

Associative Property of AdditionAssociative Property of Addition

Associative Property of MultiplicationAssociative Property of Multiplication

((a + ba + b) + ) + c = ac = a + ( + (b + cb + c) )

When three or more numbers are multiplied, When three or more numbers are multiplied, the product is the same regardless of the the product is the same regardless of the grouping of the factors. For example (2 * 3) * 4 grouping of the factors. For example (2 * 3) * 4 = 2 * (3 * 4) = 2 * (3 * 4)

Commutative Property of AdditionCommutative Property of Addition

Commutative Property of MultiplicationCommutative Property of Multiplication

When 2 numbers are added together, the sum When 2 numbers are added together, the sum is the same regardless of the order of the is the same regardless of the order of the addends. example 4 + 2 = 2 + 4 addends. example 4 + 2 = 2 + 4

When two numbers are multiplied together, the When two numbers are multiplied together, the product is the same regardless of the order of product is the same regardless of the order of the multiplicands. For example 4 * 2 = 2 * 4 the multiplicands. For example 4 * 2 = 2 * 4

Distributive Property (of Multiplication over Distributive Property (of Multiplication over Addition)Addition)

multiplying a sum by a number gives the same multiplying a sum by a number gives the same result as multiplying each addend by the result as multiplying each addend by the number and then adding the products together. number and then adding the products together.

4 × (2 + 3) = 4 × 2 + 4 × 3 4 × (2 + 3) = 4 × 2 + 4 × 3

Prop of Opposites or Inverse Property of Prop of Opposites or Inverse Property of Addition Addition

EXAMPLE:: (8 - 8 = 0) EXAMPLE:: (8 - 8 = 0) Inverse Property of Addition---- Any number Inverse Property of Addition---- Any number added to its opposite integer will always equal added to its opposite integer will always equal ZERO. If using addition the order of the ZERO. If using addition the order of the numbers doesn’t matter numbers doesn’t matter [Ex. 3 + (-3) = 0 or (-3) + 3 = 0] [Ex. 3 + (-3) = 0 or (-3) + 3 = 0]

Prop of Reciprocals or Inverse Prop. of Prop of Reciprocals or Inverse Prop. of Multiplication Multiplication

For every number, x, except zero, has a For every number, x, except zero, has a multiplicative multiplicative inverseinverse,1/x. EXAMPLE x * (1/x) = 1,1/x. EXAMPLE x * (1/x) = 1

Identity Property of Addition Identity Property of Addition

the sum of zero and any number or variable is the the sum of zero and any number or variable is the number or variable itself. number or variable itself. EXAMPLES:::: 4459907 + 0 = 4459907, - 1178 + 0 = - EXAMPLES:::: 4459907 + 0 = 4459907, - 1178 + 0 = - 1178, 1178, yy + 0 = + 0 = yy are few examples illustrating the are few examples illustrating the identity property of addition.identity property of addition.

Identity Property of Multiplication Identity Property of Multiplication

the product of 1 and any number or variable is the number or variable the product of 1 and any number or variable is the number or variable itself.itself.

EXAMPLES::::: 3 × 1 = 3, - 118997006 × 1 = - 118997006, EXAMPLES::::: 3 × 1 = 3, - 118997006 × 1 = - 118997006, ybbhybbh × 1 = × 1 = yy bbh bbh few examples illustrating the identity property of multiplication.few examples illustrating the identity property of multiplication.

Multiplicative Property of ZeroMultiplicative Property of Zero

The product of 0 and any number results in 0.That is, for The product of 0 and any number results in 0.That is, for any real number any real number a, a × 0 = 0.a, a × 0 = 0.

Closure Property of AdditionClosure Property of Addition

the sum of any two real numbers equals another real number.the sum of any two real numbers equals another real number. 2, 5 = real numbers. 2, 5 = real numbers.

2 + 5 = 7(real number).2 + 5 = 7(real number).

Closure Property of Multiplication Closure Property of Multiplication

the product of any two real numbers equals another real numberthe product of any two real numbers equals another real number.. 4, 7 = real numbers.

4 × 7 = 28(real number)

Product of Powers PropertyProduct of Powers Propertyto 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 PropertyPower of a Product Property

Power of a Power PropertyPower 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.

find the power of each factor and then multiply

Quotient of Powers PropertyQuotient 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 PropertyPower 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 PropertyZero Power Propertyany variable with a zero exponent is equal to one

example:: c°= 1

Negative Power PropertyNegative Power Property

Any variable with a negative exponent is equal to 1 over its reciprocal. ExAmPLE:: ExAmPLE:: cˉⁿ = 1/cn

Zero Product Property Zero Product Property the product of two real numbers is zero, then at least one of the the product of two real numbers is zero, then at least one of the numbers in the product (factors) must be zero. numbers in the product (factors) must be zero.

Cbr = 0 Cbr = 0 THENTHEN c = 0, b = 0, or r = 0 c = 0, b = 0, or r = 0

Product of Roots PropertyProduct of Roots Property for any nonnegative real numbers a and b.

EXAmple::: √ab = √a • √b

Quotient of Roots PropertyQuotient of Roots Property

For any nonnegative real number a and any positive real number b: eXaMpLeeXaMpLe √a/b = √a/√b √a/b = √a/√b

Root of a Power PropertyRoot of a Power PropertyI could not find this property anywhere. I could not find this property anywhere.

Power of a Root PropertyPower 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 Now you will take a quiztake a quiz!!LookLook at the at the sample problem sample problem and and givegive the the namename of the of the property illustratedproperty illustrated. .

1. a + b = b + a 1. a + b = b + a

Answer: Answer: Commutative Property (of Addition)Commutative Property (of Addition)

Now you will take a quiz!Now you will take a quiz!LookLook at the at the sample problem sample problem and and givegive the the namename of the of the property property illustratedillustrated. .

2. 4 + 2 = 2 + 4 2. 4 + 2 = 2 + 4

Answer: Answer: Community Property of Addition Community Property of Addition

11StSt power inequalities power inequalities

Single Sign Inequality ProblemsSingle Sign Inequality Problems

X is less than or equal to 2Written like: x≤2

Answer this!

How is k less than or equal to 6 written?

k≤6

ConjunctionConjunction

Two sentences combined by the word Two sentences combined by the word andand . .

EXAMPLE: xEXAMPLE: x › › 22 andand x x ‹ ‹ 11

Disjunction Disjunction

Two sentences combined by the word Two sentences combined by the word or.or.

Example:: -3 ‹ x Example:: -3 ‹ x OR OR x ‹ 4x ‹ 4

In the next slides you will review:In the next slides you will review:

Linear Linear equations equations in two in two variablesvariables

SLOPESSLOPES

Positive Slope= line running upward (rising) Positive Slope= line running upward (rising) from left to rightfrom left to right

Negative SlopeNegative Slope= lines that are falling from = lines that are falling from left to right. left to right.

Horizontal Slope=Horizontal Slope= lines that run left to right lines that run left to right (flat) with no slope (0=slope). (flat) with no slope (0=slope).

Vertical Slope= Vertical Slope= Lines that rise up and down Lines that rise up and down (undefined= slope)(undefined= slope)

SLOPE FORMULAS!SLOPE FORMULAS!

General= Ax + By + C= 0General= Ax + By + C= 0 Standard= Ax + By = CStandard= Ax + By = C Slope= rise/runSlope= rise/run Slope Formula= YSlope Formula= Y2 2 - - YY11/ X/ X2 2 - X - X11

Slope-Intercept Form= y=mx + b Slope-Intercept Form= y=mx + b (m=slope) (b= y-int.) (m=slope) (b= y-int.)

Point-Slope= y – yPoint-Slope= y – y11 = m ( x – x = m ( x – x11) )

1

Example: Graph the line y = 2x – 4.

2. Plot the y-intercept, (0, - 4).

1. The equation y = 2x – 4 is in the slope-intercept form. So, m = 2 and b = - 4.

3. slope= 2.

(1, -2) is also on the line.

1= change in y

change in xm = 2

4. Start at the point (0, 4). move 1 space to the right and 2 up to place the second point on the line.

2

x

y

5. Draw the line through these points that you just made!

(0, - 4)

(1, -2)

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 = - (x – 4) is a line

of slope m = - passing

through the point (4, 3).

1

2 1

2

(4, 3)

m = -1

2

x

y

4

4

8

8

Linear Linear SYSTEMS!SYSTEMS!

Substitution MethodSubstitution Method

StepsSteps 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 Addition/subtraction methodmethod

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, Dependent, inconsistent, consistentconsistent

FACTORINGFACTORING

The simplest method of factoring a polynomial is to factor out the greatest common factor (GCF) of each term.

Example: Factor 18x3 + 60x.

GCF = 6x18x3 + 60x = 6x (3x2) + 6x

(10)

18x3 = 2 · 3 · 3 · x · x · x

Apply the distributive law to factor the polynomial.

6x (3x2 + 10) = 6x (3x2) + 6x (10) = 18x3 + 60x

Check the answer by multiplication.

Factor each term.

Find the GCF.

60x = 2 · 2 · 3 · 5 · x

= 6x (3x2 + 10)

= (2 · 3 · x) · 3 · x · x

= (2 · 3 · x) · 2 · 5

Example: Factor 4x2 – 12x + 20.

GCF = 4.

4(x2 – 3x + 5) = 4x2 – 12x + 20Check the answer.

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)

(x + 1) (5 – y) = 5(x + 1) – y(x + 1)Check.

= 4(x2 – 3x + 5)

difference of two squares

Example: Factor x2 – 9y2.

= (x)2 – (3y)2

= (x + 3y)(x – 3y)

Write terms as perfect squares..

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 – (5y2)2

= [(x + 1) + (5y2)][(x + 1) – (5y2)]

= (x + 1 + 5y2)(x + 1 – 5y2)

x2 – 9y2

(x + 1)2 – 25y 4

2. Factor 2a2 + 3bc – 2ab – 3ac.

Some polynomials can be factored by grouping terms to produce a common binomial factor.

= 2a2 – 2ab + 3bc – 3ac

= y (2x + 3) – 2(2x + 3)

= (2a2 – 2ab) + (3bc – 3ac)

= 2a(a – b) + 3c(b – a)

= (2xy + 3y) – (4x + 6)

Group terms.

Examples: 1. Factor 2xy + 3y – 4x – 6.

Factor each pair of terms.= (2x + 3) ( y – 2) Factor out the common

binomial.

Rearrange terms.

Group terms.Factor.

= 2a(a – b) – 3c(a – b) b – a = – (a – b).

= (a – b) (2a – 3c) Factor.

2xy + 3y – 4x – 6

2a2 + 3bc – 2ab – 3ac

Factoring these trinomials is based on reversing the FOIL process.

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).

Example: Factor x2 + 3x + 2. = (x + a)(x +

b)

Express the trinomial as a product of two binomials with leading term x and unknown constant terms a and b.

= x2

F

Apply FOIL to multiply the binomials.

= x2 + (b + a) x + ba Since ab = 2 and a + b = 3, it follows that a = 1 and b = 2.

= x2 + (1 + 2) x + 1 · 2

Therefore, x2 + 3x + 2 = (x + 1)(x + 2).

O I L

+ bx

+ ax + ba

x2 + 3x + 2

Example: Factor x2 – 8x + 15.

= (x + a)(x + b)

(x – 3)(x – 5) = x2 – 5x – 3x + 15

x2 – 8x + 15 = (x – 3)(x – 5).

Therefore a + b = -8

Check:

= x2 + (a + b)x + ab

It follows that both a and b are negative.

= x2 – 8x + 15.

SumNegative Factors of 15

-3, - 5 - 8

- 1, - 15 -15

and ab = 15.

x2 – 8x + 15

Example: Factor x2 + 13x + 36. = (x + a)(x + b)

Check: (x + 4)(x + 9)

Therefore a and b are:

x2 + 13x + 36

= x2 + 9x + 4x + 36= x2 + 13x + 36.

= (x + 4)(x + 9)

= x2 + (a + b) x + ab

SumPositive Factors of 36

1, 36 37

153, 12

4, 9 13

6, 6 12

2, 18 20

x2 + 13x + 36

two positive factors of 36

whose sum is 13.

36 13

Example: Factor 4x3 – 40x2 + 100x.

A polynomial is factored completely when it is written as a product of factors that can not be factored further.

The GCF is 4x.

= 4x(x2 – 10x + 25)Use distributive property to factor out the GCF.

= 4x(x – 5)(x – 5)Factor the trinomial.

4x(x – 5)(x – 5)= 4x(x2 – 5x – 5x + 25)

= 4x(x2 – 10x + 25)

= 4x3 – 40x2 + 100x

4x3 – 40x2 + 100x

Check:

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.

1. Find the product of first and last terms

3 4 = 12

Decomposition Method

2. We need to find factors of 12whose

sum

is 8

1, 122, 63, 4

3. Rewrite the middle term decomposed into the two numbers

3x2 + 2x + 6x + 4

= x(3x + 2) + 2(3x + 2)

= (3x2 + 2x) + (6x + 4)

4. Factor by grouping in pairs

= (3x + 2) (x + 2)

3x2 + 8x + 4 = (3x + 2) (x + 2)

Example: Factor 4x2 + 8x – 5.

4x2 + 8x – 5 = (2x –1)(2x – 5)

4 5 = 20

We need to find factors of 20

whose difference is 81, 202, 104, 5

Rewrite the middle term decomposed into the two numbers

4x2 – 2x + 10x – 5

= 2x(2x – 1) + 5(2x – 1)

= (4x2 – 2x) + (10x – 5)

= (2x – 1) (2x + 5)

Factor by grouping in pairs

QUADRATIC EquationsQUADRATIC Equations

QUADRATICSQUADRATICS

A binomial expression has just two terms (usually an x A binomial expression has just two terms (usually an x term and a constant). There is no equal sign.  Its term and a constant). There is no equal sign.  Its general form is general form is ax + bax + b, where , where aa and and bb are real numbers are real numbers and a and a ≠ 0≠ 0. .

One way to multiply two binomials is to use the One way to multiply two binomials is to use the FOIL FOIL methodmethod. FOIL stands for the pairs of terms that are . FOIL stands for the pairs of terms that are multiplied: multiplied: First, Outside, Inside, LastFirst, Outside, Inside, Last..

This method works best when the two binomials are in This method works best when the two binomials are in standard form (by descending exponent, ending with standard form (by descending exponent, ending with the constant term).the constant term).

The resulting expression usually has four terms before The resulting expression usually has four terms before it is simplified. Quite often, the two middle (from the it is simplified. Quite often, the two middle (from the Outside and InsideOutside and Inside) terms can be combined. ) terms can be combined.

For example: For example:

QUADRATICSQUADRATICS

The The opposite of multiplying two binomials is to factoropposite of multiplying two binomials is to factor or break down a polynomial (many termed) expression. or break down a polynomial (many termed) expression.

Several methods for factoring are given in the text. Be Several methods for factoring are given in the text. Be persistent in factoring! persistent in factoring! It is normal to try several pairs of It is normal to try several pairs of factors, looking for the right ones.factors, looking for the right ones.

The more you work with factoring, the easier it will be to The more you work with factoring, the easier it will be to find the correct factors. find the correct factors.

Also, if you check your work by using the FOIL method, it Also, if you check your work by using the FOIL method, it is virtually impossible to get a factoring problem wrong.  is virtually impossible to get a factoring problem wrong. 

Remember!  When factoring, always take out any Remember!  When factoring, always take out any factor that is common to all the terms first.factor that is common to all the terms first.

A quadratic equation involves a single variable A quadratic equation involves a single variable with exponents no higher than 2. with exponents no higher than 2.

Its general form is where a, b, and c Its general form is where a, b, and c are real numbers and .  are real numbers and . 

For a quadratic equation it is possible to have For a quadratic equation it is possible to have two unique solutions, two repeated solutions two unique solutions, two repeated solutions (the same number twice), or no real solutions.(the same number twice), or no real solutions.

The solutions may be rational or irrational The solutions may be rational or irrational numbers.numbers.

To solve a quadratic equation, ONLY To solve a quadratic equation, ONLY IF ITS factorable:IF ITS factorable:

        1.  Make sure the equation is in the 1.  Make sure the equation is in the general form.  general form. 

        2.  Factor the equation. 2.  Factor the equation.         3.  Set each factor to zero.  3.  Set each factor to zero.          4.  Solve each simple linear equation. 4.  Solve each simple linear equation.

To solve a quadratic equation if you can’t To solve a quadratic equation if you can’t factor the equation: factor the equation:

Make sure the equation is in the general Make sure the equation is in the general form.  form. 

Identify a, b, and c. Identify a, b, and c. Substitute a, b, and c into the quadratic Substitute a, b, and c into the quadratic

formula:formula:

    Simplify. Simplify.

Tips & TricksTips & Tricks

The cool, easy thing about the The cool, easy thing about the quadratic formula is that it works on quadratic formula is that it works on any quadratic equation when put in any quadratic equation when put in the form general form.the form general form.   

When having trouble factoring a problem, When having trouble factoring a problem, the quadratic formula might be quicker. the quadratic formula might be quicker.

be sure and check your solution in the be sure and check your solution in the original quadratic equation.original quadratic equation.

Rational Rational Expressions

Expressions

Simplifying Rational Simplifying Rational ExpressionsExpressions

The objective is to be able to simplify a The objective is to be able to simplify a rational expressionrational expression

5

2x

3

92

x

x

ALWAYS, ALWAYS, ALWAYS, ALWAYS, ALWAYS Divide out ALWAYS Divide out the common factorsthe common factors

Factor the Factor the numerator and numerator and denominator and denominator and then divide the then divide the common factorscommon factors

Dividing Out Common Dividing Out Common FactorsFactors

Step 1 – Identify any factors which are common to both the numerator and the denominator.

5

5 7

x

x( )The numerator and denominator have a common factor.

The common factor is the five.

Dividing Out Common Dividing Out Common FactorsFactors

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

5

5 7

x

x( ) x

x 7

x

x 7

Factoring the Factoring the Numerator Numerator and and DenominatorDenominatorFactor the numerator.

Factor the denominator.

Divide out the common factors.

Write in simplified form.

3 9

1 2

2

3

x x

x

FactoringFactoring

Step 1: Look for common factors to both terms in the numerator.

3 9

1 2

2

3

x x

x

3 is a factor of both 3 and 9.

X is a factor of both x2 and x.

Step 2: Factor the numerator.

3 9

1 2

2

3

x x

x

3 3

12 3

x x

x

( )

FactoringFactoring

Step 3: Look for common factors to the terms in the denominator and factor.

3 9

1 2

2

3

x x

x

The denominator only has one term. The 12 and x3 can be factored.

The 12 can be factored into 3 x 4.

The x3 can be written as x • x2.

3 9

1 2

2

3

x x

x

3 3

3 4 2

x x

x x

( )

Divide and Divide and SimplifySimplify

Step 4: Divide out the common factors. In this case, the common factors divide to become 1.

3 3

3 4 2

x x

x x

( )

Step 5: Write in simplified form.

x

x

3

4 2