## Presentation on theme: "Advanced Math Chapter P"— Presentation transcript:

Review of Real Numbers and Their Properties

Natural numbers {1, 2, 3, 4, …} Advanced Math Chapter P

Whole numbers {0, 1, 2, 3, 4, …} Advanced Math Chapter P

Integers { … , -3, -2, -1, 0, 1, 2, 3, … } Advanced Math Chapter P

Rational numbers Can be written as the ratio p/q where q ≠ 0 Includes natural, whole, integers, and fractions. The decimal representation of a rational number either terminates (like 0.25) or is repeating. Advanced Math Chapter P

Irrational numbers Are not rational Have infinite non-repeating decimal representations. Advanced Math Chapter P

You try Which of the numbers above are… Natural numbers? Whole numbers? Integers? Rational numbers? Irrational numbers? Advanced Math Chapter P

Real numbers Used in everyday life to describe quantities Includes rational and irrational numbers Doesn’t include imaginary numbers Advanced Math Chapter P

Real number line Numbers to the right of origin are positive, numbers to the left are negative Nonnegative numbers are positive or zero Nonpositive numbers are negative or zero Advanced Math Chapter P

One-to-one correspondence
Between real numbers and points on the real number line Every real number corresponds to one point on the number line Every point on the number line corresponds to one real number Advanced Math Chapter P

Definition of order If a and b are real numbers, a is less than b if b – a is positive and on a number line, a is left of b Advanced Math Chapter P

Bounded intervals Have endpoints Have finite length See chart on page 3 Advanced Math Chapter P

Closed intervals Include endpoints Shown with square brackets “or equal to” Open intervals don’t include endpoints (shown with parentheses) Advanced Math Chapter P

Open intervals Don’t include endpoints Shown with parentheses Advanced Math Chapter P

Example Graph the following on a number line Advanced Math Chapter P

You try Graph the following on a number line Advanced Math Chapter P

Unbounded intervals Do not have a finite length See chart on page 4 Advanced Math Chapter P

Example Express the following using inequality notation All x in the interval (–2,4] Advanced Math Chapter P

You try Express the following using inequality notation t is at least 10 and less than 22 Advanced Math Chapter P

Absolute value Magnitude Distance between the origin and the point on the number line Advanced Math Chapter P

Properties of Absolute values
Chart on page 5 Advanced Math Chapter P

Distance between a and b

Variables Letters used to represent numbers Advanced Math Chapter P

Algebraic expressions
Combinations of letters and numbers Advanced Math Chapter P

Terms Parts of an algebraic expression separated by addition (or subtraction) Advanced Math Chapter P

Constant term Term that doesn’t contain a variable Advanced Math Chapter P

Evaluating algebraic expressions
Substitute numerical values for each of the variables in the expression Advanced Math Chapter P

You try Evaluate the following for x = 1 Advanced Math Chapter P

Substitution Principle
If a = b, then a can be replaced by b in any expression involving a. Advanced Math Chapter P

Charts Pages 6, 7, and 8 Advanced Math Chapter P

You try Exercises 98 – 104 even Advanced Math Chapter P

Factors If a, b, and c are integers such that ab = c, then a and b are factors, or divisors, of c. Advanced Math Chapter P

Prime number Integer that has exactly two factors: 1 and itself Advanced Math Chapter P

Composite Can be written as the product of two or more prime numbers Advanced Math Chapter P

Fundamental Theorem of Artihmetic
Every positive integer greater than 1 can be written as the product of prime numbers in precisely one way Prime factorization Advanced Math Chapter P

Exponential notation a to the nth power n is the exponent a is the base Advanced Math Chapter P

Properties of exponents
Chart page 12 Read first two paragraphs on page 13 Advanced Math Chapter P

Examples No calculator Advanced Math Chapter P

You try No calculator Advanced Math Chapter P

Example Rewrite with positive exponents and simplify Advanced Math Chapter P

You try Rewrite with positive exponents and simplify Advanced Math Chapter P

Scientific notation n is an integer Positive exponents mean large numbers Negative exponents mean small numbers Advanced Math Chapter P

Examples Write in scientific notation 9,460,000,000,000 Advanced Math Chapter P

You try Write in scientific notation 34,000,000 Advanced Math Chapter P

You try Write in decimal notation × 10-19 Advanced Math Chapter P

Definition of nth root Page 15 Advanced Math Chapter P

Principal nth root Page 15 Advanced Math Chapter P

Tables Page 16 Advanced Math Chapter P

Examples No calculators Advanced Math Chapter P

You try No calculators Advanced Math Chapter P

All possible factors have been removed from the radical All fractions have radical-free denominators The index of the radical is reduced Advanced Math Chapter P

Examples No calculators Advanced Math Chapter P

You try No calculators Advanced Math Chapter P

You try No calculators Advanced Math Chapter P

Rationalizing denominators
Gets rid of radical in denominator Multiply both numerator and denominator by the conjugate of the denominator Advanced Math Chapter P

You try Advanced Math Chapter P

Rationalizing numerators
Sometimes useful Not simplifying radical Multiply numerator and denominator by conjugate of numerator Advanced Math Chapter P

Rational exponents Definition page 19 Advanced Math Chapter P

You try Change from radical to rational exponent form Advanced Math Chapter P

You try Change from rational exponent form to radical form Advanced Math Chapter P

You Try Simplify: Advanced Math Chapter P

Polynomials and Special Products

Polynomial an is the leading coefficient n is the degree of the polynomial A0 is the constant term Advanced Math Chapter P

Example Coefficients are 3, 7, 8, and -5 Leading coefficient is 3 Polynomial degree 4 Advanced Math Chapter P

Polynomials in two variables
Degree of each term is sum of exponents Degree of polynomial is highest degree of its terms leading coefficient goes with highest-degree term Advanced Math Chapter P

Standard form Written with descending powers of x, then descending powers of y Advanced Math Chapter P

Add or subtract like terms (have the same variables to the same powers) by adding and subtracting their coefficients Advanced Math Chapter P

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FOIL ONLY FOR MULTIPLYING TWO BINOMIALS Product of first terms + Product of outside terms + Product of inside terms + Product of last terms Advanced Math Chapter P

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Multiplying other polynomials
Use the distributive property Add the products of each term of the first polynomial times the second polynomial Advanced Math Chapter P

Special Products Page 26 Advanced Math Chapter P

Factoring Polynomials

Factoring Writing a polynomial as a product Unless noted otherwise, you want factors with integer coefficients Completely factored when each of its factors is prime Advanced Math Chapter P

Removing a common factor
Distributive property in reverse First step in factoring a polynomial Advanced Math Chapter P

You try Advanced Math Chapter P

Factoring special polynomial forms
Page 34 Come from special product forms in section P.3 Advanced Math Chapter P

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Trinomials with binomial factors
FOIL in reverse May involve trial and error Advanced Math Chapter P

You try Advanced Math Chapter P

Factoring by Grouping Sometimes works for polynomials with more than three terms Sometimes several different options will work Can use to factor trinomials Advanced Math Chapter P

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Domain The set of real numbers for which an algebraic expression is defined Usually all real numbers, except any that make the expression equal an imaginary number Or make it undefined Advanced Math Chapter P

Examples Find the domain of the following: Advanced Math Chapter P

You try Find the domains of the following: Advanced Math Chapter P

Simplifying Rational expressions
Factor each polynomial completely Divide out common factors List the domain by the simplified expression The domain of the simplified expression cannot include numbers that weren’t in the domain of the original expression Advanced Math Chapter P

Examples Write the following in simplest form Advanced Math Chapter P

You try Write the following in simplest form Advanced Math Chapter P

Operations with rational expressions
Factor Then multiply, divide, add, or subtract using the rules for fractions Advanced Math Chapter P

You try Advanced Math Chapter P

Complex fractions Have separate fractions in the numerator, denominator, or both. Two ways to solve Making one fraction in numerator and one in denominator and dividing Multiply numerator and denominator by LCD of all fractions involved Advanced Math Chapter P

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Simplifying expressions with negative exponents
Factor out the common factor with the smaller exponent When factoring, subtract exponents Advanced Math Chapter P

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Difference Quotient Have a difference on the top and a constant or degree 1 term on the bottom In calculus, sometimes you have to rewrite them by rationalizing the numerator so that the expression is defined. Advanced Math Chapter P

Errors and the Algebra of Calculus

Common algebraic errors

Algebra of Calculus Sometimes writing things in an “unsimplified” way makes doing calculus operations easier Read on your own Let me know if you have questions Advanced Math Chapter P

Example Simplify the expression Advanced Math Chapter P

Example Write the fraction as the sum of three terms Advanced Math Chapter P

You try Write the fraction as the sum of three terms Advanced Math Chapter P

Cartesian Plane Rectangular coordinate system Named after René Descartes Ordered pair: (x, y) Horizontal x-axis Vertical y-axis Origin: where axes intersect Advanced Math Chapter P

Scatter plots Each point is plotted Dots are not connected Advanced Math Chapter P

Distance formula Pythagorean theorem d Advanced Math Chapter P

You try Find the distance between (-3, 2) and (3, -2) Advanced Math Chapter P

Verifying a right triangle
Showing that three given points are vertices of a right triangle. Plot the points Use the distance formula to find the distances between the points. See if the distances work in the Pythagorean theorem Advanced Math Chapter P

Example Use the distance formula to show that the points (9,4), (9,1), and (-1,1) form a right triangle. Advanced Math Chapter P

Midpoint formula To find the midpoint of the line segment joining two points, average the x-coordinates and average the y-coordinates. Midpoint has coordinates Advanced Math Chapter P

Example Find the midpoint of the segment joining the points (1, 1) and (9, 7). Advanced Math Chapter P