Merced College; Don Power

INTERMEDIATE ALGEBRA - CH 5, LECTURE

5.1 Integer Exponents and Scientific Notation

Exponents - you know basic definition;

exponent form vs expanded form for 3⁴

Identify the base and exponent.

Ex: Is (a+b)² = a² + b² ?

Definition is?

What is missing if you expand (a+b)(a+b) by FOIL?

In general, if you can't remember algebra rules, try substitution (let a=2, b=3...)

Exponents

Properties:

1. Product: a^r * a^s = a^r+s When multiplying factors with equal bases, add the exponents

Why? Exponents count factors. So 2³ * 2⁴ = (2*2*2)*(2*2*2*2) = 2⁷

Don't change the base

2. Product-to-Power: (ab)^r = a^rb^r Distribute exponents to every factor

(i.e. across multiplication/division); Ex: (2x⁵)³

Why? Again, consider expanded form: (2x⁵)³ = 2x⁵*2x⁵*2x⁵ = 2³x¹⁵ = 8x¹⁵

Note: This includes numerical factors

3. Power-to-Power: (a^r)^s = a^rs To raise a power to a power, multiply exponents

(don't change the base)

Why? Exponents count factors. So (2³)⁴ = 2³*2³*2³*2³ = 2¹² by property 1

4. Quotient: a^r / a^s = a^r-s This corresponds to property 1

In practice, this can lead to negative exponents: Ex: x³ / x⁸

Except for scientific notation, it's better to avoid a negative exponents (think cancellation):
you should leave a positive power in the denominator: Ex: x³ / x⁸ = 1 / x⁵

For each base ask:

How many factors are there in the numerator?

How many factors are there in the denominator?

How many of those factors get cancelled?

Finally, How many of those factors remain, and where are they (top or bottom)?

5. Quotient-to-Power: (a/b)^r = a^r / b^r (2x / y⁵)³ = (2x)³ / (y⁵)³ = (by rules 2,3) 2³x³ / y¹⁵ = 8x³ / y¹⁵

Special exponents 0 and 1, negative exponents:

6. a¹ = a and a⁰ = 1 (provided a¹0) Isn't 2³ / 2³ = 1? With subt of exponents we have 2⁰ = 1.

7. (a/b)^-r = (b/a)^r Neg exponents correspond to reciprocals

Why? Since a⁰ = 1, we get: 1 / a^r = a⁰ / a^r = (by rule 6) a^0−r = a^−r

In particular:

a^−r / 1 = 1 / a^r and 1 / a^-r = a^r / 1

Implication: To clear a negative exponent, move the factor across the fraction bar

My suggested order of ops for clearing up exponents:

1. Distribute exponents to factors in parentheses (including numerical factors)

Caution: not across terms: if we have + or - in the parenthesis, we can't distribute the exponent.

Ex: (1+1)² ≠ 1² + 1²

2. Move factors with negative exponents

3. Combine factors with like bases:

Add exponents if on same side of fraction bar

Subtract exponents if on opposite sides of fraction bar

4. Evaluate exponents for numerical factors

Examples: ,

Scientific Notation

Tasks:

1. Translate back and forth, for both large and small numbers

2. Enter into calculator and read calculator display:

Entry: Use key marked E, EE, EXP, not e or e^x For 8.0 x 10⁻⁵, type 8 EE (−) 5

Reading: Exponent of 10 follows E or is shown separated (or maybe raised)

3. Multiply and divide using scientific notation

Manually. Issue: translate result into scientific notation

Ex: 8E-5 / (4E6 * 5E-8) = 0.4E−3 = 4E−4

With calculator

Ex:

5.2 Adding and Subtracting Polynomials

Set up, if necessary; clear parentheses and combine like terms.

I suggest underlining like terms, if needed, instead of introducing extra parentheses.

Ex: Subtract (-4x²+5x-3) - (2x²+x-7)

Vertical format is good for longer problems

Position function for a free-falling object on the earth's surface: units feet, seconds, ft/sec

h(t) = –16t² + v₀t +s₀

For our applications, we'll substitute values for t: i.e., find h(2) etc.

5.3 Multiplying Polynomials

General technique: multiply every term in first polynomial times every term in second polynomial

Multiply coefficients, copy bases, add exponents for like bases

Justification: distributive law

Ex: Multiply, using FOIL: (3x+2)(x-7)

Vertical format is sometimes useful: x³ - 3x² + 4x - 3 times 4x² + 3x -7

Special products:

Square of a binomial

Know the formula

Learn to do by FOIL, without writing out

Ex: (2x-3)²

Product of a Sum and Diff [of Same Two Terms] yields a difference of squares

Know the formula

Avoid having to do FOIL

Ex: (2x-3)(2x+3)

Contrast the last two cases:

Remember which pattern contains a middle term, and remember how to calculate it

Application: Calculate area of a side, or volume, if sides of rectangular solid are n, n+1, n+2

Application: Calculate revenue R = xp where price per unit is 50–0.001x

Then find R(100) (what does this mean?)

#109-110 relate to the general compound interest formula

#111-112 illustrate multiplication rules using geometry

5.4 Factoring: Grouping and Special Products

General Procedure for Factoring

1. First step: factor out GCF

Ex: Factor 6x³ - 12xy + 18xz + 6x [common factor]

Ex. Factor 3x(x-4)+y(x-4)-z(x-4) [common factor]

2. Count terms

4 terms (or more): factor by grouping (this lesson)

3 terms : "trinomial factoring" (lesson 5.5)

2 terms: special forms (this lesson)

difference of squares

sum or difference of cubes

3. Check to see if you can factor further

Factoring by Grouping

Possibilities

Usually: Grouping terms in pairs, factor out common factors

Ex: Factor x³y³ + 2x³ + 5x²y³ + 10x²

Ex: Factor 15x³ + 12x² – 10x + 8

Less often: Combinations involving special factoring (hold these for later):

a. Group terms in pairs, with one pair involving a difference of squares

Ex: Factor x² – 4y² – x + 2y

Ex: Factor y² −16x² + 2y + 8x

b. Grouping 3 terms as be a perfect square;

that perfect square minus a constant could be a difference of squares

Ex: Factor x² – 6x + 9 – 4y²

Ex: Factor x²-8x+16-25y²

Ex: Factor x²+4xy+4y²-16

Special Factoring

Perfect squares: you get them immediately if you follow the procedure in 5.5 below

Ex: Factor 32x² -96xy +72y² [always factor out common factor first]

Differences of squares: you may have to factor out common factor to see the pattern

Ex: Factor 98x²-32y²

Clearing fractions if they exist: How do you do factor out LCD when there is a fraction?

Ex: Factor LCD is 15, so write Distribute 15 to clear the fractions.

Sum/ difference of cubes:

Formula: u³ ± v³ = (u ± v) (u² –/+ uv + v²)

We have a short, easy factor and a longer, harder one. Notice how to construct the long one.

[Note: We say a²+ab+b² cannot be factored. We mean it cannot be factored using rational numbers. It does factor as a+sqrt(ab)+b)(a-sqrt(ab)+b) -- but this is not over rationals]

Ex (easy): a³ - 8

Ex (hard):

Ex (even harder):

In general, for difference of 6th powers: do as diff of squares first. Ex: x⁶ – 64

5.5 Factoring Trinomials

Factoring trinomials with leading coeff of 1

Key to getting the numbers: Factor the const; add those factors to get middle coeff.

Ex: Factor x²-5xy-24y²

Factor x²-14xa+48a²

Handling case with -x² Ex 15-2x-x²

Either (5 x)(3 x) and fill in signs, or

Factor out "-" to start with

Def: prime polynomial Ex: x²+4, x²-6 (Why in each case?)

Factoring other trinomials by trial and error

Ex: 18a²+3ab-28b²

The first terms of the factors have to be factors of the first term - consider all factor pairs

Factor pairs of 18 are 1,18 or 2,9 or 3,6

The last terms of the factors have to be factors of the last term

Factor pairs of 28 are 1,28 or 2,14 or 4,7

Check each trial by calculating possibilities for the middle term:

Specifically, write down the outer and inner products

If testing (3a 4b)(6a 7b), these products are 21ab and 24ab

Decide on the signs:

From the last term, do we need the same or opposite signs? Here, negative, so opposite signs

Since we need +3ab for the middle term, we need the signs -21ab and +24ab

Find where to insert the - and + into the two factors

(3a - 4b)(6a + 7b)

Mult out by FOIL to verify the solution:

Suggested Strategy:

1. Factor out any common factor

2. Write all factor pairs of the first and last coefficients

3. The first term to factor is the one with the fewest factor pairs;

The second term to factor is the one with the most factor pairs.

4. Start with factors that are closest together (from both sets of factor pairs).

5. Write out the inner and outer products.

Consider: Do we need the same sign or opposite signs?

Can we add/subtract the inner and outer products to get the middle term?

6. Try swapping the second terms in the two parentheses before moving on to the next factor pair.

7. Don't check arrangements that would introduce common factors

This will reduce the nr of trials to be tested!

Additional trinomial examples

20x²+44x-15

12x²-67x+35

Alternate procedure, splitting the middle term ("delayed grouping")

Ex: 12x²-67x+35 [remember to factor out the common factor first -- otherwise...]

This technique can confirm that a polynomial is prime (i.e. not factorable)

Disadvantage: finding all factor pairs if the product ac is large (as with this example)

Extension examples:

Factor the x⁶ into x³ times x³, etc: cut the exponent in half

Fractional coefficients: either

a. Factor the numerator and denominator separately, or

b. Factor out the LCD from the entire expression

Factoring and area

Additional sample problems:

Factor 4x² -2x - 30

4x²-14xa+48a² Common factor only

Show that 2x² -7x + 15 is prime, by both the trial-and-error and the alternate procedures.

5.6 Solving Equations by Factoring

Quadratic equation: One of the form ax² + bx + c = 0

If we can factor, we can solve.

We may need to rearrange to set everything equal to 0

Repeated solutions: We may get the same solution twice (if the quadratic is a perfect square)

Square root principle: If a² = b, then a = ±sqrt(b) a and b may be any algebraic expressions.

Ex: Solve x² = 13

Solving Non-linear Equations by Factoring (including quadratic equations)

Zero-factor property: For any real numbers a and b, a*b = 0 implies a=0 or b=0 (or both)

If a product a*b = 0 then a = 0 or b = 0

a and b may be numbers, variables, or algebraic expressions

The same property also applies to three or more factors.

Application: any equation that can be factored, after being set equal to 0.

Particular application: quadratic equations (can be written in form ax² + bx + c = 0):

Cases: 1. 8x² = 32 (missing x term)

Divide out common numerical factor of 8: becomes diff of squares.

2. 5x² = 15x (missing constant) Factor out common factor 5x.

Do not divide out common variable factor If we divide by x, we may be dividing by 0.

3. x² + x = 6 (no missing terms; factorable)

4. x² +2x +7 = 0 (not factorable; other techniques are available to solve -- later in course)

Applications

Number problem (consecutive integers): # 96, but also accept negative integers.

Geometry: # 100

Free-falling object: # 106

Break-even analysis: # 110

Recap from chapter 2:

Strategy for linear and quadratic equations

0. Floating step: collect like terms whenever possible

1. Clear grouping symbols: usually by applying the distributive law

Ex: x(x² + 4x -5) = 20

2. Clear fractions (optional unless you have a variable in the denominator): multiply by the LCD.

Ex (fractions): Solve 4x/5 + 5/4 = 3x/10-1/8

Ex (decimals): 0.3x +0.16 = 0.25x

Ex (variable in denominator): 5x + 1/3 = 2/x

3. Linear or quadratic? (i.e. first degree or second degree?)

If quadratic, set everything equal to 0 and factor to get two linear equations.

This also works for 3rd or higher degree polynomials

See the discussion of the zero-factor property in this lesson.

4. Move terms with the var to one side of the equation, move other terms to the other side

First, identify which terms are out of place.

Then for each misplaced term,

Either:

a. Put the opposite of the misplaced term on both sides of the equation; it can go

below the like terms on both sides, or

in-line on both sides (text's technique), or

b. Move the misplaced terms across the "=" sign and change signs, or

c. Add/subtract mentally

5. Collect like terms: If variables are involved, factor out the variable so it appears only once.

6. Divide [or multiply by the reciprocal if you didn't clear fractions]

7. Check by substitution

McKeague

5.1 Rational Exponents

Motivation: diagonals of rectangles can be found by square roots

Def of -- "square root of x"

1. If then r is a real number such that r² = x

2. r is the prinicpal root (i.e. the positive root [for even roots])

So the negative square root of x is -

Square roots of negative numbers are not defined (for real numbers)

-- so both x and r are positive only

Def of higher roots

Cube root of x: (real number r such that r³ = x) r pos or neg Ex = -3

Fourth root of x: (real number r such that r⁴ = x) r and x are pos only Ex = 2

"nth root of x:" (real nr r such that rⁿ = x);

General restriction: for even roots [i.e. n is even], both x and r can only be positive.

Negative radicands result in:

neg root if radicand is odd Ex: cube root (-125)

no real number if radicand is even Ex: fourth root (-16)

Terminology: radical expression, index, radical sign, radicand

Rational numbers as exponents

For a rational (fractional) exponent,

Numerator = power

Denominator = root

See the theorem

Illustration: Solve x = 8 ^ (1/3) by cubing both sides; then solve x = 8 ^ (2/3)

Variables under a radical

Basic definition: If n is even, then

If n is odd, then

If we know that the variable x is nonnegative, then: Key property:

If we assume all vars are nonnegative, then we can divide exponent by the index to simplify

(Restriction is necessary for even indices, not odd indices)

Ex: 5th root of (2⁵x¹⁵y⁸z¹⁷w⁴⁴)

If we rewrite this using rational exponents, and convert to mixed numbers (or divide)

Whole number is exponent outside the radical.

Numerator is exponent inside the radical.

or, if we divide:

Quotient is exponent outside the radical

Remainder is exponent inside the radical

Review properties of exponents -- these all apply to rational exponents, too.

1. (a^r)(a^s) = a^r+s 6. a^r / a^s = a^r-s so rule 4. 1/a^s = a^-s = a^-s/1 and 1/a^-r = a^r / 1 = a^r

3. (ab)^r = a^rb^r 5. (a/b)^r = a^r / b^r

2. (a^r)^s = a^rs

Thm 5.1: provided a, n positive

App: constructing a golden rectangle from a square of side 2

5.2 More Expressions Involving Rational Exponents

Multiplication with terms with rational exponents

Distributing monomial across binomial (Exercise 8)

FOIL (Exercise 13)

Perfect square (Exercise 20)

Mult sum and difference to get difference of squares (Exercise 30)

Dividing by a single term (Exercise 46)

Factoring

Factoring out the GCF (Exercise 50)

Text doesn't require student to discover the GCF, but it's useful to know this:

The GCF is each base with its lowest exponent

For each factor, result of factoring is same as for dividing

(so we factor by subtracting exponents)

Example (actual simplification problem from calculus):

Calculus note: This is the derivative of

Factoring expressions quadratic "in form"

"like" trinomials (Exercise 58)

this works if one exponent is half of the other -- let u = variable and smaller exponent

"like" differences of squares (Exercise 60)

Fraction addition and subtraction, where LCD includes rational exponent (Exercise 68)

Application: Formula for annual rate of return on investment

Derivation (not in book): A = P(1-r)^t → r = (A/P)^1/t -1

Example: Exercise 78: Investment of $800 grows to $1600 after 5 years.

5.3 Simplified Form for Radicals

Standards

1. No perfect squares under sq root symbol -- (so, factor out perfect squares)

Ex:

No perfect cubes under cube root symbol, etc -- (so, factor out perfect cubes, etc)

Ex:

So...no perfect nth powers under nth root symbol (so, factor out perfect nth powers)

Ex:

2. No fractions under a radical -- so, rationalize the denom (as below)

3. No radicals in a denominator -- so, rationalize the denom (as below)

Ex. for class: for n = 2, 3, 4, 5

Removing factors from the radicand:

Either: (1) Factor out a perfect square (for sqrt), perfect cube (for cube roots), etc., or

(2) Write the prime factorization and divide each exponent by the index

The quotient is the exponent outside the radical

The remainder is the exponent inside the radical.

or, if it doesn't divide evenly, factor out just enough to get even division (factor out base ^remainder.)

Rationalizing the denom: Multiply as necessary to make the radicand a perfect square (for sqrt), a perfect cube (for cube roots), a perfect 4th power (for 4th roots), etc.

One technique:

Write the prime factorization of each factor in the radicand

Goal is to mult as necessary to get exponent to be a multiple of the index

(For each factor, subtract its exponent from the next highest multiple of the index)

5.4 Addition and Subtraction of Radical Expressions

Def of similar radicals: have the same index and the same radicand

Technique: simplify first, then collect like terms (i.e. similar radicals)

We can factor out the common radical factor

Examples:

Like #11, 12, 14 together:

#16

#26

Like #28