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Chapter P

Chapter P. Prerequisites: Fundamental Concepts of Algebra. P.1 Algebraic Expressions & Real Numbers. Objectives. Evaluate algebraic expressions Use mathematical models Find the intersection of 2 sets Find the union of 2 sets Recognize subsets of the real numbers Use Inequality symbols

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Chapter P

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  1. Chapter P Prerequisites: Fundamental Concepts of Algebra

  2. P.1 Algebraic Expressions & Real Numbers

  3. Objectives • Evaluate algebraic expressions • Use mathematical models • Find the intersection of 2 sets • Find the union of 2 sets • Recognize subsets of the real numbers • Use Inequality symbols • Evaluate absolute value • Use absolute value to express distance • Identify properties of the real numbers • Simplify algebraic expressions

  4. Intersection of Sets • What they have in common • A = {all tall children} • B = {all girls} • A intersect B = {all tall girls} • All children that are girls AND are tall

  5. Union of Sets • Combination of everything in both sets • A = {all tall children} • B = {all girls} • A union B = {all girls OR tall children} = {all girls and all tall boys}

  6. A group of biology majors are taking Biology I & Chem. I. A group of chemistry majors are taking Calculus, Chem. I and Physics I. The Physics majors enrolled in Calculus, Physics I, and Chem I. What is the intersection of the 3 groups? • Students in biology, chemistry, & physics. • Students in chemistry. • Students in calculus. • Students in physics.

  7. Absolute Value • │x│ represents the distance between x and zero • Distance is always a positive quantity, therefore going left or right x units results in a distance of x units • │x - 2│ represents the distance between x and 2 • Distance is again always positive. (i.e. the distance between 2 and 3 is 1 and the distance between 2 and 1 is 1) │3 - 2│ = │1 - 2│ = 1

  8. Real numbers are a field • Commutative (addition & multiplication) • Associative (addition & multiplication) • Identity (additive = 0 & multiplicative = 1) • Inverse (additive = -x & multiplicative = 1/x) • Distributive (multiplication over addition) • ALL these properties are useful when manipulating algebraic expressions & equations

  9. P.2 • Exponents and Scientific Notation

  10. Objectives • Use the product and quotient rules • Use the zero-exponential rule • Use the negative-exponent rule • Use the power rule • Find the power of a product • Find the power of a quotient • Simplify exponential expressions • Use scientific notation

  11. Product & Quotient Rules

  12. Example • Simplify:

  13. Quotient Rule explainszero-exponent rule • Any real number divided by itself (except 0) equals 1 • If x is any nonzero number & y is an exponent:

  14. Quotient Rule Explains Negative Exponent Rule

  15. Working with Negative Exponents • In general, expressions are not considered simplified when negative exponents are present. • A negative exponent in the denominator becomes positive when moved to the numerator • A negative exponent in the numerator becomes positive when moved to the denominator

  16. Raising an Exponent to an Exponent (Power Rule) • Exponents are multiplied • WHY?

  17. When is your expression simplified? • No negative exponents are present • Each base appears only once • No parentheses remain • Example:

  18. Scientific Notation • What is it? A number greater than or equal to 1 & less than 10 (either pos. or neg.) multiplied by 10 raised to an exponent • Example:

  19. Why Use Scientific Notation? • It allows us to express very large numbers or very small numbers in a more concise manner. • Diminishes the error in writing very large or small numbers by eliminating the need to have all zeros written. (easy to have one too many or too few zeros)

  20. Converting into Scientific Notation

  21. Rules of Thumb • Count decimal places you move to place the decimal to the right of one non-zero digit • Large numbers are represented by multiplying by ten raised to a positive exponent • Small numbers are represented by multiplying by ten raised to a negative exponent

  22. P.3 • Radicals & Rational Exponents

  23. Objectives • Evaluate square roots • Simplify (nth root of nth power) • Use product & quotient rules to simplify square roots • Add & subtract square roots • Rationalize denominators • Evaluate & perform operations with higher roots • Understand & use rational exponents

  24. Principal Square Root • It is true that 4 squared and (-4) squared both equal 16, BUT the principal square root of 16 is 4 NOT -4 • By convention, the radical symbol represents the positive (or PRINCIPAL) square roots of a number, thus for real numbers, x, greater than or equal to 0:

  25. Examples

  26. Multiplying & Dividing with Radicals (Roots) • A product or quotient under a radical can be written as the product or quotient of separate radicals • Products or quotients involving square roots can be expressed as a single square root involving products or quotients under the radical

  27. Adding & Subtracting Square Roots • ONLY when you’re taking the square root of the same number can you add or subtract square roots

  28. Simplify Expressions, then Add/Subtract (if possible)

  29. What is a conjugate? • Pairs of expressions that involve the sum & the difference of two terms • The conjugate of a+b is a-b • Why are we interested in conjugates? • When working with terms that involve square roots, the radicals are eliminated when multiplying conjugates

  30. Multiplying conjugates which involve square roots

  31. Expressions with radicals in the denominator are NOT simplified • Eliminate the radical from the denominator by multiplying by the numerator and the denominator by the conjugate of the denominator • Sometimes the result may not LOOK simpler!

  32. Other Roots • The nth root of a number means “what number could you raise to the nth power to get your original number?” • You can take an odd root of a negative number or a positive number. • You can only take an even root of a positive number.

  33. Rules for other roots • Add and subtract only same roots of same number (i.e. you can add cube roots of 3 but NOT cube roots of 3 and cube roots of 4) • Multiply & divide same roots following same rules as square roots

  34. Subtract, if possible, & simplify:

  35. Expressing roots as rational exponents • Any root can be expressed as a rational exponent, then rules of exponents apply

  36. Expressions may involve exponents AND roots • If possible, it’s often easier to take the root first (the rational exponent), then raise the value to the other exponent

  37. P.4 • Polynomials

  38. Objectives • Understand the vocabulary of polynomials • Add & Subtract polynomials • Multiply polynomials • Use FOIL in polynomial multiplication • Use special products in polynomial multiplication • Perform operations with polynomials in several variables

  39. A polynomial in x is many terms added or subtracted with each term involving a constant and x raised to a power. • Only same powers of x can be added/subtracted • When multiplying polynomials, the distributive property holds. (i.e. every term in one polynomial must be multiplied by every term in the other polynomial.

  40. Special Products

  41. P.5 • Factoring Polynomials

  42. Objectives • Factor out the greatest common factor • Factor by grouping • Factor trinomials • Factor difference of squares • Factor perfect square trinomials • Factor sum & difference of cubes • Use a general strategy for factoring • Factor expressions containing fractional & negative exponents

  43. Factoring strategies • FIRST: Look for greatest common factor • Group terms (if 4 or more) to find common terms between groups • If only 3 terms, rewrite into 4 terms by multiplying leading coefficient by the constant term (a times c), then rewrite bx as the sum of 2 terms whose product of their coefficients is ac (then group as in previous item)

  44. EXAMPLE

  45. Factor by Recognition • Difference of Squares • Difference or Sum of Cubes • Opposite signs cause all middle terms to cancel out

  46. Factor Completely

  47. P.6 • Rational Expressions

  48. Objectives • Specify domain of a rational expression • Simplify rational expressions • Multiply rational expressions • Divide rational expressions • Add & subtract rational expressions • Simplify complex rational expressions • Simplify fractional expressions that occur in calculus • Rationalize numerators

  49. Domain restrictions • No values can be substituted in for x that would create a zero denominator or a negative value under a positive root

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