Math 72 Course Pack - Modesto Junior College



Math 72 Course Pack

Fall 2010

Section 0249 ONLY

Instructor: Yolande Petersen

DO NOT BUY THESE NOTES IF YOU HAVE A DIFFERENT INSTRUCTOR

Inside:

• Lecture Notes Outline with writing space for your notes

• Syllabus

• Calendar of Material Coverage and Exam Dates

• Homework Assignments for each section

• Cumulative Review Problems (sample test problems, Petersen style only)

• Final Exam Review

Suggestions for use

The lecture notes pages are in reverse order, upside down, and punched on the "wrong" side for a reason! My notes read like a book, with the printed side on the left and handwritten extra notes on the right. If you want your notes look like mine, insert the lecture pages into a binder with the blank back of page 1 on top. When you turn the first page, page 1 will be on your left, with the blank back of page 2 on your right, where you can write additional notes. Effort was made to minimize the number of pages printed to reduce your cost, while leaving enough space for your notes to be arranged in an orderly way. If you don't like this arrangement, feel free to order the pages however you like. The supplements (syllabus, homework assignments, and review problems) can be inserted into a binder in the "usual" way.

Mrs. Petersen's website:

Before you take this class, you may find it helpful to read the document at the link, "Teaching Style and Educational Philosophy" to decide whether this instructor is a good match for you.

Chapter 5 Review Concepts

Distributive Property (multiplying polynomials by monomials)

Ex a

FOIL (multiplying 2 binomials) – First, Outer, Inner, Last

Special Product Formulas

1. (f + s)2 = f2 + 2fs + s2

2. (f – s)2 = f2 – 2fs + s2

3. (f + s)(f – s) = f2 – s2

Common Errors:

6.1 The Greatest Common Factor (GCF); Factoring by Grouping

Factors - things multiplied to make a product

It is sometimes desirable to write things as factors (pieces)

Prime factored form - a product of factors where none of the factors can be "broken down" further (with exponents consolidated)

Greatest Common Factor (GCF) – the largest factor (small broken-down piece) included in all the numbers

Least Common Multiple (LCM) - the smallest multiple (large multiplier) that includes all the numbers

|Finding the GCF |Finding the LCM |

|Write all the numbers in prime factored form |Write all the numbers in prime factored form |

|For each factor, choose the smallest exponent common to all the |For each factor choose the largest exponent possible |

|numbers |Write the product |

|Write the product | |

Factoring out the GCF – the reverse of the distributive law

1. Find the GCF (by eyeballing or previous method) and write it in front

2. Divide the GCF out of each term

3. Write the "leftovers" inside parentheses

Factoring by Grouping – take out identical "clumps" of stuff

6.2 Factoring Trinomials (of form x2 + bx + c) – reverse of FOIL

x2 + bx + c factors to (x + ?)(x + ?)

How do we know whether it's + or - , and how do we get ?

Some examples of FOIL

Factored Unfactored

(x + 1)(x + 2) = x2 + x + 2x + 2 = x2 + 3x + 2

(x – 3)(x – 5) = x2 – 5x – 3x + 15 = x2 – 8x + 15

(x – 4)(x + 3) = x2 + 3x – 4x – 12 = x2 – x – 12

(x – 2)(x + 5) = x2 + 5x – 2x – 10 = x2 + 3x – 10

Observations on signs

1. If c is positive [pic]

a. If b is positive [pic]

b. If b is negative [pic]

2. If c is negative [pic]

a. If b is positive [pic]

b. If b is negative [pic]

Observations on numbers

1. c is the product of the 2 binomial numbers [pic]

2. If the 2 binomial numbers have the same sign, b

3. If the 2 binomial numbers have different signs, b

6.3 More Factoring Trinomials (ax2 + bx + c)

ax2 + bx + c factors to (?x + ?)(?x + ?)

Factoring by grouping – long process, no guesswork

1. Multiply[pic]

2. For c positive, find 2 factors whose sum = b

For c negative, find 2 factors whose difference = b

3. Rewrite the whole polynomial with the middle term written as the sum or diff.

4. Factor by grouping

Ex a

Trial and Error

• For simple numbers with few factors, it's much faster than grouping

• For numbers with many factors, requires much guesswork and luck

• Rules for signs are same as section 6.2

• Rules for numbers require "mix and match" to get the middle number, b

6.4 Special Factoring

Special formulas (same as before, written in reverse):

1. f2 – s2 = (f + s)(f – s)

2. f2 + 2fs + s2 = (f + s)2

3. f2 – 2fs + s2 = (f – s)2

Ask: Does my binomial or trinomial fit one of these 3 patterns

Binomials – Use 1st formula for squares, last 2 formulas for cubes

Trinomials - Use formulas 2 or 3

More Formulas – cubes

4. f3 + s3 = (f + s)(f2 – fs + s2)

5. f3 – s3 = (f – s)(f2 + fs + s2)

Common Errors:

x3 + y3 [pic] (x + y)3

x3 – y3 [pic] (x – y)3

Note: x2 – y2 can be factored

x2 + y2 can't be factored

Factoring Summary

1. Can a GCF be factored out? (6.1)

2. 2 terms: Is it (f2 – s2) or (f3 + s3) or (f3 – s3)? (6.4)

3. 3 terms: Is it (f2 + 2fs + s2) or (f2 – 2fs + s2)? (6.4)

Does trinomial [pic] 2 binomial factoring work easily? (6.2)

Is there a number in front of x2 for trinomial [pic] 2 binomials? (6.3)

4. 4 terms: Does factoring by grouping work?

5. Can any factors be further factored?

6.5 Solving Quadratic Equations – Factoring

Quadratic equation – 2nd degree

Standard form: ax2 + bx + c = 0

Zero Factor Property – basis for solving quadratic equations

If ab = 0, then a = 0 or b = 0 (one of the numbers MUST be zero)

Ex a

Procedure for solving

1. Write equation in standard form (get 0 on one side)

2. Factor

3. Set each piece (factor) equal to 0

4. Solve each piece; check

The number of solutions is equal to or less than the degree of the equation.

How what is the maximum number of solutions of a quadratic equation?

How many solutions could a cubic equation potentially have?

6.6 Applications of Quadratic Equations

Rectangles: A = LW; P = 2L + 2W

Triangles:

Pythagorean Theorem: In a right triangle with hypotenuse c and legs a & b,

a2 + b2 = c2

Ex The hypotenuse of a right triangle is 1 cm longer than the longer leg of the triangle. The shorter leg is 7 cm shorter than the longer leg. Find the length of the longer leg.

Numbers

Velocity/Distance

Ex The height of a ball with an initial velocity of 128 ft/sec which travels t seconds is described by the equation:

h = 128t – 16t2

a) What is the height of the ball after 2 seconds?

b) For what values of t is h = 0?

c) Interpret the values from b) in real life terms.

d) For what values of t is h = 240 ft?

e) After how many seconds do you think the ball will reach its peak?

7.1 The Fundamental Property of Rational Expressions (Canceling Fractions)

Rational Expression: Can be written as P/Q, where P and Q are polynomials , Q[pic]0

(Note: Q = 1, so non-fractions can be made into fractions)

P/Q is undefined when Q = 0.

A place where the denominator is undefined is called a

Ex a For what values is the expression [pic] undefined?

Lowest Terms – The rational expression P/Q is in lowest terms if there are no common factors in the denominator.

Fundamental Property of Rational Expressions (Canceling Rule)

[pic] where K [pic] 0

7.2 Multiplying and Dividing Rational Expressions

Multiplying: [pic]

Smart Way: Cancel (reduce) as much as possible before multiplying

Procedure:

1. Factor

2. Cancel and rewrite the remaining factors

3. Note which bad points have disappeared (for extra credit)

Dividing: [pic]

7.3 The Least Common Denominator (LCD)

Recall: The LCD is the least common multiple of the denominators. It is as large as or larger than each of the individual denominators.

Finding the LCD of Polynomial Factors

1. Factor each denominator completely, writing repeat factors in exponent form

2. Write each unique factor to the highest power possible

Caution: Don't confuse the LCD with the GCF.

Ex a

Building a fraction to match a denominator

1. Compare the new and old denominators, and find "what's missing" from the old denominator. When in doubt, divide the LCD by the old denominator.

2. Multiply top and bottom by the missing factor.

7.4 Adding and Subtracting Rational Expressions

Rules: [pic] and [pic] for Q[pic]0

Caution: 2 fractions must have same denominator. If not, a common denominator (LCD) must be used.

Procedure -- Adding/Subtracting with same denominator

1. Add or subtract numerators, keep denominator

2. Factor and/or cancel if possible

Adding/Subtracting with different denominators

1. Find the LCD

2. Build the fractions to match LCD

3. Add or subtract as above

7.5 Complex Fractions

• Typically have 4 layers (2 top, 2 bottom)

• We can treat 4 layers as 2 separate fractions, divided

2 Solving methods

Method 1: Invert bottom fraction and multiply

Method 2: Multiply both fractions by LCD to clear 2 denominators

Method 1 is generally more reliable; Method 2 can be fast but is prone to errors

Clearly Separated Layers – Use Method 1

Ex a

Partial Layers – Use either method

7.6 Equations with Rational Expressions

Recall:

|Expressions |Equations |

|no equal sign |has equal sign |

|Goal: Keep denominator by putting numerator over LCD |Goal: get rid of denominator by multiply by LCD |

|Final answer may have mixed variables and numbers |Final answer: x = number (variable is isolated) |

|Process is to simplify |Process is to solve |

Solving Procedure

1. Find LCD

2. Multiply both sides by LCD

3. Cancel factors to get rid of denominators

4. Solve

5. ***Check if solution is a bad point – This is now required, not optional

Solving for a Specified Variable

1. Clear denominators if necessary

2. Get all terms with desired variable on one side, all other terms on other side

3. Factor out desired variable

4. Divide by "junk"

7.7 Applications

Distance/Rate/Time

d = rt; isolating for different variables, r = d/t or t = d/r

Work/Rate/Time

w = rt or r = w/t or t = w/r

"3 hours to paint a room" can be interpreted in 2 ways:

1. t = 3 hours

2. r = 1 room/3 hours or 1/3 room/hour (this is most commonly used)

8.1 Evaluating Roots

square rooting - the reverse process of squaring

Suppose we are given x2 = 25. What is the "unsquare" of 25?

All square roots of "a"

If "a" is a positive, real number,

[pic] is the positive square root of "a"

- [pic] is the negative square root of "a"

Note the difference between word form and symbol form:

1. "the square roots of x"

2. [pic]

Types of Roots

1. Rational Roots – If "a" is a perfect square, then [pic] is rational, e.g.

2. Irrational Roots – If "a" is positive but not a perfect square, [pic] is irrational

To get a ballpark idea of a number:

3. Non-real roots – If "a" is negative, [pic] is non-real

Pythagorean theorem

c2 = a2 + b2 or [pic]

Higher roots – It's helpful to know some perfect squares, cubes, 4th powers, etc.

|x |1 |2 |

|1. Factoring |fast, simple |Doesn't solve every equation |

|2. Square Root |fast, simple |Doesn't solve every equation |

|3. Complete the Square |Solves every equation |Requires thinking |

|4. Quadratic Formula |Solves every equation |Tedious, requires many steps |

Square Root Method – works when b – 0 (only have x2 and constant terms, no x term)

Square Root Property

If a2 = k, then a = + [pic] or a = –[pic]

Remember: Quadratics may have

1. 2 solutions [pic] x2 = positive number (or (stuff)2 = positive #)

2. no solution [pic] x2 = negative number

3. one solution [pic] x2 = 0

9.2 Completing the Square

Suppose we have x2 + 10x + 25 = 36

Procedure for Completing the Square

1. Get equation in the form x2 + bx = c (x2 and x terms on left, number on right)

2. Find the needed number to complete the square

nn = [pic]

3. Add it to both sides.

4. Factor the perfect square. Write it as a square that looks like (x + b/2)2

5. Square root both sides

6. Solve for x

9.3 Quadratic Formula

Recall:

Quadratic equations have the form ax2 + bx + c = 0

Quadratic equations have 2, 1, or 0 solutions

Quadratic Formula

The equation ax2 + bx + c = 0 has solutions:

x = [pic] or x = [pic]

Putting the 2 together:

x1,2 = [pic]

Your book derives this formula on p. 608. We won't!

The inside of the square root is sometimes called the discriminant.

• If b2 – 4ac > 0 [pic] 2 real roots

• If b2 – 4ac = 0 [pic] 1 real root

• If b2 – 4ac < 0 [pic] no real root

9.4 Complex Numbers

The number i is defined as i = [pic] (a non-real number)

i2 = -1

Complex numbers have the form

a + bi e.g. -3 + 2i or 7 - 4i

There are pure real, pure imaginary numbers, and numbers with a mix of each.

Imaginary numbers refers to any number with an imaginary part (either pure imaginary or a mix of real & imaginary parts).

Complex numbers include all 3 types (pure real, pure imaginary, mixed)

Adding and Subtracting – add/subtract real and imaginary parts separately, similar to like terms

Dividing

• Imaginary numbers are not allowed in the denominator

• Rationalize by multiplying top and bottom by the conjugate, similar to the way radicals in the denominator are eliminated.

Multiplying imaginary conjugates produces a real number:

(a + bi)(a - bi) = a2 – abi + abi – b2i2 = a2 – b2(-1) = a2 + b2

9.5 Graphing Quadratic Equations/Quadratic Functions

Recall some features of linear functions (1st degree):

1. They can be written as y = mx + b

2. They are straight lines

3. Important landmarks are m (slope) and b (y-intercept)

Some features of quadratic equations (2nd degree):

1. They can be written as y = ax2 + bx + c

2. They are

3. Important landmarks include:

Direction of opening

Vertex and axis of symmetry

Width (of parabola)

Finding the Vertex (better way than trial & error)

x-coordinate: x = [pic][pic]

y-coordinate: plug the value of x into the original equation to get y

Procedure for Graphing a 2nd degree equation (parabola)

Decide if the parabola opens up or down

Calculate the vertex

Plot at least 1 extra point to find the width of the parabola

Intercepts of a Graph of a Quadratic Function

A graph has its x-intercepts when y = f(x) = 0 (set equation = 0)

The x-values are the places where the graph crosses the line.

[pic]

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