Algebra Notes

Honors Algebra Notes

Substitution – use to simplify expressions or when working with formulas.

Variable – symbols for unknown quantities. Establish your variable for the quantity you know least about.    

Interest = Principle ´ rate ´ time

Order of Operations

-Parentheses

-Exponents

-Multiplication

-Division

-Addition

-Subtraction

Domain - the set of “x” only.

Range - the set of “y” only.

Steps to Problem Solving

1.      Read the problem

2.      Establish your variable

3.      Set up an equation using the variable

4.      Solve the equation

5.      Answer the question and check for logic

Absolute value – the distance from “n” to zero.

Rule of Signs in Multiplication and Division:

1.      An even number of negatives = a positive

2.      An odd number of negatives = a negative

Closure properties – the sum of any two real numbers is also real numbers.

Commutative properties- the order in which you add or multiply two numbers does not affect the result.

Associative Properties- When you add or multiply any three real numbers, the grouping of the numbers does not affect the results.

Consecutive integers – one number after another in a certain pattern.

a)      Even- x+2, x+4, x+6…

b)      Odd- x+2, x+4, x+6…

Reciprocals- two numbers whose product equals one.

Identity (as a solution) – Infinitely many solutions. Ex. 8 = 8

No Solution- a = { }, a = q.    Ex. 8 = 6

Steps for solving equations with variables on both sides

1. Simplify each side
2. Combine all variables so that they are all on one side.
3. Solve for that variable.

Charts to Organize Data:

The equation will come from the total column.

Exponents:

In the term, 2x², 2 is the Coefficient, x is the Base, and ² is the Exponent

Exponent-short hand for repeated multiplication

Terms-Numbers separated by + or – signs.

Polynomials:

Polynomial - expressions with multiple terms.

To Add/Subtract Polynomials:

Like Terms:

1.      Identical Bases

2.      Identical Powers

 Combine by adding the coefficients only. Ex: 3x³ + 5x³ = 8x³

1.      When adding like terms, add only coefficients

2.      When multiplying, add powers of like bases.

3.      When powers have powers, multiply the powers.

Multiplying Polynomials:

(3x+1)(2x-5) = 6x²-13x-5

First: 3x ´ 2x = 6x²

Outside: 3x ´ -5 = -15x

Inside: 1 ´ 2x = 2x

Last: 1 ´  -5 = -5

Motion Problems

Rectangle – Area = length ´ width

Uniform – the same all the way around all four sides.

Factor – A number that can be divided evenly into a given number.

In the division of monomials, subtract powers of like bases.

Any number to the power of zero must equal one.

Prime – does not factor.

A binomial to the power of two will produce a trinomial.

Factoring Methods:

1.      GCF

2.      Binomial – Difference of squares

3.      Trinomial – T-charts

4.      4 term polynomial - grouping

Factoring by Grouping:

3 v. 1 or 2 v. 2 only if it is a perfect square.

Ex. 2ab-6ac + 3b-9c  = 2a(b-3c) + 3(b-3c) = (b-3c)(2a+3)

Volume = length ´ width ´ height

 Steps to Add or Subtract Fractions:

1. LCD- Factor all denominators
2. Combine
3. Factor
4. Simplify

Steps to Add or Subtract Mixed Expressions:

1. Put non-fractions over one
2. Find LCD (factor first)
3. Combine like terms
4. Simplify or factor

Ratio – a comparison of two or more numbers

1. Convert units when possible
2. Reduce to lowest terms
3. No units in the final answer

Proportions – two or more equal ratios.

To Solve Equations with Fractions:

1. Multiply the LCD to both sides
2. Cross-cancel all denominators
3. Solve

Mixture Problems:

Chemistry-

            Dilute a concentration by adding water [0% chemical]

            Strengthen a concentration by adding more chemical [100% chemical]

AB + 0 = C (A-X)

Nuts-

AB+CX = D (A+X)

Coins-

5N+10D+25Q = X

Percent of Change:

Change in Value       x

Original Value     =100

Simple Interest- I=P´ r´ t

Work Problems:

(1/a + 1/b) T = 1

 X-coordinate – Abscissa

Y-coordinate – Ordinate

0 – origin

Slope Intercept of a line - y = mx + b

            y= slope intercept

            m=slope (steepness of a line)

            b= y-intercept

            (y)-(y)        up

m=       (x)-(x)   = over

Standard Form of a Line – Ax + By = C

1. x+y together
2. no fractions
3. x term positive

Parallel lines have equal slopes.

Function – a relationship between the Domain and Range such that each x has a unique y.

Quadratic Function – ax² + bx + c = y

Quadratic term Linear term       Constant term

Solution of a Parabola:

1. Write the equation out in Standard Form
2. Axis of Symmetry: x = -b/2a
3. Graph the axis of symmetry
4. Substitute the axis of symmetry into the equation as x to find the vertex. If “a” is negative, the vertex will be a maximum point, if it is positive the vertex will be a minimum point.
5. Substitute two x values greater than or less than the axis value to find the y values. Graph these values and connect them to form a parabola.

Variation:

1.      Direct: divide

2.      Inverse: multiply

System – two or more related equations using the same variables(can be graphed on the same coordinate plane).

To Solve Systems of Equations:

1. Substitution
2. Addition

Wind and Water Problems:

Age Problems:

1. Establish variable expressions for present.
2. Establish variable expressions for past or future based on present expression.
3. Make an equation from these expressions.

Inequalities:

1. < >: Open circle           £ ³ : Closed circle
2. Whether you multiply or divide by a negative; you must reverse the inequality symbol.

 Graphing Linear Inequalities:

1. < >: Dotted line
2. £ ³ : Solid line
3. < £: Shade below
4. > ³: Shade above

Rational – any number that can be written as a fraction.

Irrational – any number that cannot be written as a fraction.

Restriction on Roots:

1. You can never have a negative radical.
2. To guarantee a positive result, use absolute value bars whenever you go from a even power to an odd power.

Pythagorean Theorem: a²+b²=c²

The Quadratic Formula: -b ± ¶b²-4ac

                                              2a

The Discriminate: b²-4ac

1. b²-4ac < 0        no real roots
2. b²-4ac = 0        one real root
3. b²-4ac > 0        two real roots