Intermediate AlgebraOptional Pre-Final Exam Review

1 – Basic Algebra Review 2 – Graphs & Equations of Lines 3 – Solving Systems of Equations 4 – Inequalities 5 – Polynomials & Factoring 6 – Rational Expressions & Functions 7 – Radical Expressions & Functions 8 – Quadratic Functions 9 – Exponents & Logarithms 10 – Conic Sections appearwill

Exercises

1 – Basic Algebra Review

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1 – Basic Algebra Review Rules for Order of Operations To make sure an expression is always evaluated in

the same way by different people, the Order of Operations convention was defined

Mnemonic: “Please Excuse My Dear Aunt Sally”

Parentheses Exponents Multiply/Divide

Add/Subtract

Always: Evaluate & Eliminate the innermost grouping first

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1 – Basic Algebra Review

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2 – Graphs & Equations of Lines Plotting Points aka Graphing Points

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2 – Graphs & Equations of Lines Solutions to Equations Any point on a graphed equation is a Solution

2 – Graphs & Equations of Lines What is Slope & Why is it Important?

Using any 2 points on a straight line will compute to the same slope.

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andif )4,5()5,2(

2 – Graphs & Equations of Lines The Slope-Intercept Form of a Line

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int

2048

x

ym

xy

Remember the CoverUp Method?

Slopes of Parallel Lines m1 = m2

But wait! can you be sure that it’s not the same line ?

Slopes of Perpendicular Lines m1 = -1 / m2

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31 yxandxy

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3 – Solving Systems of EquationsUsing the Substitution Method

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yx

yx

B

A

3 – Solving Systems of Equations Using the Elimination (Addition) Method

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7

yx

yx

B

A

3 – Solving Systems of Equations Solution to 3 Equations

Adding (A) and (C) will eliminate y (A) 2x – y + 3z = 6 (C) 2x + y + z = -2

(D) 4x +4z = 4 first new equation in 2 variables

Adding (B) and 5·(C) will also eliminate y

(B) 3x – 5y +4z = 75·(C) 10x + 5y + 5z = -10

(E) 13x + 9z = -3 second new equation in 2 variables

Solve (D) and (E) like a system of two equations (next page) Use Substitution or Addition

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632

zyx

zyx

zyx

C

B

A

Well use substitution of x from (D) into (E) to find z (D) 4x +4z = 4

(D1) x = 1 - z move 4z to the other side, divide by 4

Substitute x from (D1) into (E)

(E) 13x + 9z = -3 13(1 – z) + 9z = -3

13 – 13z + 9z = -3 use distribution, then simplify -4z = -16 z = 4

Substitute z into (D) or (E) or (D1) to find x (D) 4x + 4(4) = 4

4x + 16 = 4 4x = -12 x = -3

Substitute x and z into (A) or (B) or (C) to find y (C) 2(-3) + y + (4) = 4

-6 + y + 4 = 4 y = 6 Solution is (-3, 6, 4)

Solution to (D) 4x + 4z = 4

Continued (E) 13x + 9z = -3

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7453

632

zyx

zyx

zyx

C

B

A

4 – Inequalities Intersections, Unions & Compound Inequalities

Set Diagrams Intersections of Sets Conjunctions of Sentences and Unions of Sets Disjunctions of Sentences or Interval Notation Domains

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4 – Inequalities Expressing Domains With Interval Notation

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4 – Inequalities Using the Absolute Value Principle

|x + 1| = 2 x + 1 = 2 or x + 1 = -2 |2y – 6| = 0 2y – 6 = 0 |5x – 3| = -2 no solution

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4 – Inequalities When an equation has 2 absolute values?1|92||3| xx

5 – Polynomials & Factoring Subtracting Polynomials

To subtract polynomials, add the opposite of the second polynomial.

(7x3 + 2x + 4) – (5x3 – 4) add the opposite!(7x + 2x + 4) + (-5x3 + 4)

Use either horizontal or vertical addition. Sometimes the problem is posed as subtraction:

x2 + 5x +6 make it addition x2 + 5x +6 - (x2 + 2x) _ of the opposite -x2 – 2x__ 3x +6

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5 – Polynomials & Factoring Multiplying Two Polynomials

To multiply a polynomial by a polynomial, we use the distributive property repeatedly.

Horizontal Method:(2a + b)(3a – 2b) = 2a(3a – 2b) + b(3a – 2b) = 6a2 – 4ab + 3ab – 2b2 = 6a2 –ab – 2b2

Vertical Method: 3x2 + 2x – 5 4x + 2

6x2 + 4x – 10 12x3 + 8x2 – 20x____

12x3 + 14x2 – 16x – 10

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5 – Polynomials & Factoring FOIL: Used to Multiply Two Binomials )53)(25( yxyx

5 – Polynomials & Factoring Find the Greatest Common Factor 7a – 21 =

7(? – ?) = 7(a – 3)

19x3 + 3x = x(? + ?) = x(19x2 + 3)

18y3 – 12y2 + 6y = 6y(? – ? + ?) = 6y(3y2 – 2y + 1)

xyyxyx 493514 223

5 – Polynomials & Factoring Factor by Grouping 8t3 + 2t2 – 12t – 3 2t2(4t + 1) – 3(4t + 1) (4t + 1)(2t2 – 3)

4x3 – 6x2 – 6x + 9 2x2(2x – 3) – 3(2x – 3) (2x – 3)(2x2 – 3)

y4 – 2y3 – 12y – 3 y3(y – 2) – 3(4y – 1) Oops – not factorable via grouping

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5 – Polynomials & Factoring Using a Factor Table - Trial & Error

Let’s use x2 + 13x + 36 as an example Factors must both be sums: (x + ?)(x + ?) Pairs=c=36 Sum=b=13 1, 36 37 2, 18 20 3, 12 15 4, 9 13 ok quit!

x2 + 13x + 36 = (x + 4)(x + 9)

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The ac Grouping Method: ax2 + bx + cSplit bx into 2 Terms: Use a Table based on a·c

Let’s use 3x2 – 10x – 8 as an example ac = 3(-8) = -24

One factor is positive, the other negative and larger. Pairs=ac=-24 Sum=b=-10 1, -24 -23 2, -12 -10 quit! 3, -8 -5 4, -6 -2

3x2 – 10x – 8 = 3x2 + 2x – 12x – 8 = split the middle x(3x + 2) – 4(3x + 2) = do grouping = (3x + 2)(x – 4)

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5 – Polynomials & Factoring Factoring Perfect Square Trinomials?

x2 + 8x + 16 = (x + 4)2

(x)2 (4)2 2(x)(4) = 8x yes, it matches

t2 – 12t + 4 = not a PST (t)2 (-2)2 2(t)(-2) = -4t no, it’s not -12t

25 + y2 + 10y = (y + 5)2

y2 + 10y + 25 descending order (y)2 (5)2 2(y)(5) = 10y yes, it matches

3x2 – 15x + 27 = not a PST 3(x2 – 5x + 9) remove common factor (x)2 (-3)2 2(x)(-3) = -6x no, it’s not -5x

PST Tests:

1. Descending Order

2. Common Factors

3. 1st and 3rd Terms (A)2 and (B)2

4. Middle Term 2AB or -2AB

5 – Polynomials & Factoring Difference of Squares Binomials Remember that the middle term disappears? (A + B)(A – B) = A2 - B2

It’s easy factoring when you find binomials of this pattern A2 – B2 = (A + B)(A – B)

Examples: x2 – 9 = (x)2 – (3)2 = (x + 3)(x – 3) 4t2 – 49 = (2t)2 – (7)2 = (2t + 7)(2t – 7) a2 – 25b2 = two variables squared (a)2 – (5b)2 = (a + 5b)(a – 5b) 18 – 2y4 = constant 1st, variable square 2nd 2 [ (3)2 – (y2)2 ] = 2(3 + y2)(3 – y2)

5 - Factoring the Difference between 2 CubesX3 – Y3 = (X – Y)(X2 + XY + Y2)

F3 – L3 factors easily to (F – L)(F2 + FL +L2) Examine 27a3 – 64b3

(3a)3 – (4b)3 (3a – 4b)(9a2 + 12ab + 16b2) Remember to remove common factors and to factor completely

p3 – 8 2x6 – 128 = 2[x6 – 64] (p)3 – (2)3 2[(x2)3 – 43] (p – 2)(p2 + 2p + 4) 2(x2 – 4)(x4 + 4x2 + 16) 2(x + 2)(x – 2)(x4 + 4x2 + 16)

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5 - Factoring the Sum of 2 Cubes X3 + Y3 = (X + Y)(X2 – XY + Y2)

F3 + L3 factors easily to (F + L)(F2 – FL +L2) Examine 27a3 + 64b3

(3a)3 + (4b)3 (3a + 4b)(9a2 – 12ab + 16b2) Remember to remove common factors and to factor completely

p3 + 8 2x6 + 128 = 2[x6 + 64] (p)3 + (2)3 2[(x2)3 + 43] (p + 2)(p2 – 2p + 4) 2(x2 + 4)(x4 – 4x2 + 16)

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5 - Definition

Principle of Zero Factors

5 - Solving a Quadratic Equation by Factoring 028102 23 xxx

solve

6 – Rational Expressions & Functions Multiplying Fractions

(Use parentheses for clarity)Factor expressions,

then cancel like factors

6 – Rational Expressions & Functions Dividing Fractions

Change Divide to Multiply by Reciprocal,follow multiply procedure

6 - Finding the LCD (must be done before adding or subtracting 2 or more RE’s)

1. Factor each denominator completely into primes.

2. List all factors of each denominator. (use powers when duplicate factors exist)

3. The LCD is the product of each factor to its highest power.

28z3 = (22) (7)(z3)

21z = (3)(7)(z)

LCD=(22)(3)(7)(z3)

a2 – 25 = (a + 5)(a – 5)

a + 7a + 10 = (a + 5) (a + 2)

LCD = (a + 5)(a – 5)(a + 2)

1. Find the LCD.2. Express each rational

expression with a denominator that is the LCD.

3. Add (or subtract) the resulting rational expressions.

4. Simplify the result if possible.

6 - Adding or subtracting rational expressions with unlike denominators

6 - Simplifying Complex Rational Ex’s Method 1: Multiplying by 1

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6 - Simplifying Complex Rational Ex’s Method 2: Multiplying by Reciprocal Making the top and bottom into single

expressions, then multiplying by reciprocal.

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6 – Rational Expressions & Functions Rational Equations: False Solutions

Solve a Rational Equation by Multiplying BOTH SIDES by the LCD

Warning: Clearing an equation may add a False Solution A False Solution is one that causes a divide by zero situation in

the original equation Before even starting to solve a rational equation, we need to

identify values to be excluded What values need to be excluded for these?

t ≠ 0 a ≠ ±5 x ≠ 0

6 – Rational Expressions & Functions Clearing & Solving a Rational Equation

What gets excluded? x ≠ 0What’s the LCD? 15xWhat’s the solution?

6 – Rational Expressions & Functions Dividing a Polynomial by a Polynomial

Use the long-division process

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7 – Radical Expressions & Functions Examples to Simplify

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7 – Radical Expressions & Functions Practice

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Express using rational exponents:

Simplify using rational exponents:

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3

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3

8

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13

1 3

Rational Expressions Where the Numerator is greater than 1

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2 42 323

2221616

62342

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3

22216

Using Exponent Arithmetic, it’s a little easier

7 – Radical Expressions & Functions The Product Rule for Radicals

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333 33 43 3025322532240

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7 – Radical Expressions & Functions The Quotient Rule for Radicals

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7 – Radical Expressions & Functions A Radical Expression is Simplified When:

1. Each factor in the radicand is to a power less than the index of the radical

2. The radicand contains no fractions or negative numbers

3. No radicals appear in the denominator of a fraction

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7 – Radical Expressions & Functions Definitions A Radical Equation must have at least one

radicand containing a variable

The Power Rule: If we raise two equal quantities to the same power,

the results are also two equal quantities If x = y then xn = yn

Warning: These are NOT equivalent Equations!

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7 – Radical Expressions & Functions Equations Containing One Radical To eliminate the radical,

raise both sides to the index of the radical

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x

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Check

Equations Containing Two Radicals Make sure radicals are on opposite sides Sometimes you need to repeat the process

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7 – Radical Expressions & Functions i, The Basis of the Complex Number System

7 – Radical Expressions & Functions Powers of i

1

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3

2

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ii

1

11

8

7

6

5

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8 – Quadratic Functions The Square Root Principle

Solve by factoring

x2 – 16 = 0 (x+4)(x-4)=0 x=4,-4

Then by the square root property

x2 – 16 = 0 x2 = 16 x=4,-4

08811 2 x

8 – Quadratic Functions Using Completing the Square to Solve an Equation

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8 – Quadratic Functions Introducing … The Quadratic Formula!

The Quadratic Formula is used to find solutions to any quadratic equation

The formula was derived using completing the square and the square root property.

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4

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4

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8 – Quadratic Functions Solving Quadratic Form Equations The variable u is often used to replace squares

of variables or expressions

ixandxxandx

uxSolveNowuandu

uusolvenowuu

rewriteandxuletxx

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2

2

2

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8 – Quadratic Functions (parabolas) Graph f(x) = (x – 3)2

Maximums & Minimums382 2 xxykhxaxf 2)()(

9 – Exponents & Logarithms Graphing an Exponential Function

9 – Exponents & Logarithms Translating Right and Left

9 – Exponents & Logarithms General Definition of Logarithms

3125log2100log

32log

02:0log

510

2

2

impossiblealso

solutionnoimpossible y

10 - Introduction to Conic Sections Parabola Circle Ellipse Hyperbola

10 – Conic SectionsRemember Parabolas? Two styles: Functions & Relations

Find the Vertex:

x = -b/(2a),

(or y = -b/(2a))

solve for y or x

10 – Conic SectionsA Circle has a Center and a Radius

Find the center & radius

An Ellipse also has a Center and Foci

What Next? Getting a Good Grade on the Final!Thank you all for

a good class.

MrV