Int. Alg. Final Exam Review Sheet December 2007 Page 24 of 25

Section R.3: Operations on Signed Numbers

To find the least common denominator:

  1. Factor each denominator.
  2. Write down the common factor(s), and then copy any additional factors.
  3. Multiply those numbers; the product is the least common denominator.
  4. Convert each rational number to an equivalent fraction with the LCD as denominator.

The Distributive Property of Real Numbers:

If a, b, and c are real numbers, then

Section R.4: Order of Operations

Exponents

Repeated multiplication can be written more efficiently using the notation of exponents:

Many calculators and computer languages use the “caret” symbol to denote exponents:

Order of Operations:

  1. Evaluate expressions within parentheses first. Begin with the innermost parentheses and work outward.
  2. Evaluate expressions involving exponents next, working from left to right.
  3. Evaluate expressions involving multiplication and division next, working from left to right.
  4. Evaluate expressions involving addition and subtraction next, working from left to right.

Section R.5: Algebraic Expressions

Simplifying Algebraic Expressions:

·  by Combining Like Terms:

·  by removing parentheses using the distributive property:


Section 1.1: Linear Equations

We write an equation when we know what we want the answer to a multi-step to calculation to turn out to be.

  1. Key fact that allows us to solve equations: Properties of Equality (i.e., the equality remains true if you do the same thing to both sides of the equation.): An equation can be transformed into an equivalent equation by adding or subtracting the same quantity on both sides of the equation, or by multiplying or dividing both sides of the equation by the same nonzero quantity.
  2. The goal in manipulating equations when we solve them is to isolate the variable on one side of the equation.
  3. When solving equations, proceed opposite the order of operations:
  4. Simplify both sides of the equation as much as possible by combining like terms and removing parentheses suing the distributive property.
  5. Look for something to add or subtract to both sides.
  6. Look for something to multiply or divide both sides.
  7. Look for an exponent to raise both sides to.

Section 1.2:An Introduction to Problem Solving

English / Math
is, was, are, yields, equals, gives, results in, is equal to, is equivalent to / =
product of, of (with a fraction or %) / ´
sum of / +
difference / -
x more than y / x + y
x less than y / y - x
increase / +
decrease / -
twice x or 2 times x / 2x
n times x / nx
Two consecutive integers / n, n+1

Area of a rectangle: A = Width ´ Length

Perimeter = sum of the lengths of the sides of a shape

Rate ´ Time = Amount

Distance = rate ´ time

Mixture Problems: Portion from Item A + Portion from Item B = Total

Percentage problems:

Problem Solving with Mathematical Models

  1. Identify what you are looking for.
  2. Give names to the unknowns.
  3. Translate the problem into the language of mathematics. Use pictures to help you when possible.
    Solve the equation.
  4. Check the reasonableness of your answer.
  5. Answer the original question.

Section 1.3: Using Formulas to Solve Problems

Some Geometric Formulas
Shape / Formula
Square / Area
Perimeter
Rectangle / Area
Perimeter
Triangle / Area
Perimeter
Trapezoid / Area
Perimeter
Parallelogram / Area
Perimeter
Circle / Area
Circumference
Cube / Volume
Surface Area
Rectangular Solid / Volume
Surface Area
Sphere / Volume
Surface Area
Right Circular Cylinder / Volume
Surface Area
Cone / Volume


Section 1.4: Linear Inequalities

Properties of Inequalities: An inequality can be transformed into an equivalent inequality by adding or subtracting any quantity to both sides, or multiplying or dividing by any positive quantity. If both sides are multiplied or divided by a negative quantity, then the inequality symbol gets reversed.

Section 1.5: Compound Inequalities

Intersection of two sets: all the elements in both sets.

Union of two sets: only the elements common to both sets.

Compound Inequalities Using and: the solution is the intersection of the two sets.

Compound Inequalities Using or: the solution is the union of the two sets.

Section 1.6: Absolute Value Equations and Inequalities

To solve absolute value equations, use the facts that:

è is equivalent to u = a or u = -a.

è is equivalent to u = v or u = -v.

To solve absolute value inequalities, use the facts that:

è is equivalent to

è is equivalent to

è is equivalent to or

è is equivalent to or

Section 2.1: Rectangular Coordinates and Graphs of Equations

To graph a linear equation by plotting points, make a table with two rows of values, plot the points, then draw the line.

Graph
Table:
x / y
0 / -5
2 / -1
/ Plot points:
/ Draw line:


To graph a nonlinear equation by plotting points, make a table with many rows of values, plot the points, then draw a smooth curve connecting the points.

Graph y = 0.5x2 + 1.
x / y
-3 / 5.5
-2 / 3.0
-1 / 1.5
0 / 1.0
1 / 1.5
2 / 3.0
3 / 5.5
/

An x-intercept of a graph is the x-coordinate of a point on the graph that crosses or touches the x-axis.

A y-intercept of a graph is the y-coordinate of a point on the graph that crosses or touches the y-axis.

This graph has x-intercepts at x = -2, x = -1, x = +1, and x = +2, and a y-intercept at y = 4.

Section 2.2: Relations

A relation is a “link” from elements of one set to elements of another set.

If x and y are two elements in these set, and if a relation exists between x and y, then we say:
x corresponds to y,
or y depends on x,
and we write x ® y.
We may also write a relation where y depends on x as an ordered pair (x, y).

The domain of a relation is the set of all inputs to the relation.

The range of a relation is the set of all outputs of the relation.

Section 2.3: An Introduction to Functions

Section 2.4: Functions and Their Graphs


Section 3.1: Linear Equations and Linear Functions

Definition: Linear Equation

A linear equation (in two variables) is an equation of the form

Ax + By = C

Where A, B, and C are real numbers and A and B both cannot be zero. This is called the standard form for the equation of a line.

To graph a linear equation using intercepts:

  1. Let y = 0 and solve for x (for the x-intercept)
  2. Let x = 0 and solve for y (for the y-intercept)

Equation of a Vertical Line
A vertical line is given by an equation of the form
x = a
Where a is the x-intercept.
Equation of a Horizontal Line
A horizontal line is given by an equation of the form
y = b
Where b is the y-intercept. /

Section 3.2: Slope and Equations of Lines

Definition: Slope

The slope m between two points with coordinates (x1, y1) and (x2, y2) is defined by the formula

.

If x1 = x2, then the line between the two points is a vertical line, and the slope m is undefined.

Properties of Slope:

m > 0 è the line slants upward from left to right

m < 0 è the line slants downward from left to right

m = 0 è the line slants is horizontal

m = undefined è the line is vertical

Point-Slope Form of a Line

If you know the slope of a line (m) and one point on that line (x1, y1), then the equation of the line is given by the equation

(Point-Slope Form of a Line)

An equation of a line L with slope m and y-intercept b has a slope-intercept form of

.

Equation of a line from two points:

1)  Find the slope.

2)  Use the point-slope form.

Section 3.3: Parallel and Perpendicular Lines

Definition: Relationship between the slopes of parallel lines

Two nonvertical lines are parallel if and only if their slopes are equal and they have different y-intercepts. Vertical lines are parallel if they have different x-intercepts.

Definition: Relationship between the slopes of perpendicular lines

Two nonvertical lines are perpendicular if and only if the product of their slopes is -1. Alternatively, their slopes are negative reciprocals of one another. Vertical lines are perpendicular to horizontal lines.

Section 3.4: Linear Inequalities in Two Variables

è Step 1: Write the inequality as an equality, then graph the equation using a dashed line if it is a strict inequality (< or >), or using a solid line if it is not a strict inequality (£ or ³).

è Step 2: Pick a test point and see if the ordered pair satisfies the inequality. If it does satisfy, shade the half of the plane on the side of the line containing the point. If it does not satisfy, shade the other half.

Section 3.5: Building Linear Models

You should be able to write equations for linear models given a verbal description, a direct variation, or a set of data.

Section 4.1: Systems of Linear Equations in Two Variables

Solution by substitution:

  1. Solve one equation for one of the variables.
  2. Substitute that expression for that variable into the other equation.
  3. Solve for the remaining variable.
  4. Back-substitute into the first equation to get the value for the other variable.

Section 4.2: Problem Solving: Systems of Two Linear Equations Containing Two Unknowns

You should be able to write out a model for a real-world situation involving two equations in two unknowns, then solve that system to get an answer.


Section 4.3: Systems of Linear Equations in Three Variables

Example of a system of three equations in three unknowns that is in triangular form:

Notice: the name triangular form comes from the “blank” triangular space in the lower left corner due to no x or y variables. Also, this system is really easy to solve using back-substitution:

Steps for Solving a System of Three Linear Equations in Three Unknowns Using Elimination

  1. Eliminate the same one variable from two of the equations using elimination.
  2. Use elimination to remove a second variable from those two equations.
  3. Solve for the remaining variable.
  4. Substitute the answer for that variable into the remaining equations.

Section 4.4: Using Matrices to Solve Systems

Example of a system of three equations in three unknowns that is in triangular form:

è

Use row operations to convert the matrix into triangular form.

è è è


Section 4.5: Determinants and Cramer's Rule

Definition: Determinant of a 2 ´ 2 matrix

Suppose that a, b, c, and d are real numbers. The determinant of the 2 ´ 2 matrix , written as , is .

Cramer’s Rule for solving a system of two linear equations in two unknowns

The solution to the system of equations is given by

and

Provided that

Definition: Determinant of a 3 ´ 3 matrix

The determinant of the 3 ´ 3 matrix , written as , is calculated using the determinants of 2 ´ 2 matrices as follows:

.

Cramer’s Rule for solving a system of three linear equations in three unknowns

The solution to the system of equations
with

, , , and

is given by

, , and .

Section 4.6: Systems of Linear Inequalities

Graph the system . /

Section 5.1: Adding and Subtracting Polynomials

To add polynomials, combine like terms.

To subtract polynomials, combine like terms.

A polynomial function is a function whose rule is a polynomial. The domain of a polynomial function is all real numbers. The degree of a polynomial function is the value of the largest exponent on the variable.

Section 5.2: Multiplying Polynomials

Extended form of the Distributive Property:

To multiply polynomials, use distribution, or multiply each term in the first polynomial by each term in the second polynomial.

Example: (x2 + 3x + 9)(x2 – 2x +3) = x4 –2x3 +3x2

+3x3 -6x2 +9x

+9x2 -18x +27

=x4 +x3 +6x2 -9x +27

Section 5.3: Dividing Polynomials; Synthetic Division

Dividing a polynomial by a monomial: Divide the monomial into each term of the polynomial, and cancel when possible.

Dividing a polynomial by a polynomial using long division:

Long division of polynomials is a lot like long division of numbers:

  1. Arrange divisor and dividend around the dividing symbol, and be sure to write them in descending order of powers with all terms explicitly stated.
  2. Divide leading terms, then multiply and subtract.
  3. Repeat until a remainder of order less than the divisor is obtained.

Example:

è

Dividing a polynomial by a polynomial using synthetic division: THIS IS A SHORTCUT THAT ONLY WORKS WHEN THE DEGREE OF THE DIVISOR IS 1 (I.E., THE DIVISOR IS x – c) !!!

Synthetic division is a shorthand way to divide a polynomial by a linear factor.:

  1. Write c outside the bar and the coefficients of the dividend inside the bar.
  2. Bring the leading coefficient straight down.
  3. Compute c times the number in the bottom row, and write the answer in the middle row to the right.
  4. Add and repeat until all coefficients are used up.

The Remainder Theorem

If the polynomial P(x) is divided by x – c, then the remainder is the value P(c).

The Factor Theorem

If P is a polynomial function, then x – c is a factor of P if and only if P(c) = 0.

(This can be used to see if a divisor divides evenly into a dividend quickly)

Section 5.4: Greatest Common Factor; Factoring by Grouping