Rational Expressions Rational Expressions are fractions (ratios) that contain polynomial expressions in the numerator and denominator. Examples:
Rational Expressions Simplifying Rational Expressions with Monomial Numerators and Denominators Method 1: Use laws of exponents Method 2: Expand and divide out common factors
Rational Expressions Simplifying Rational Expressions with Polynomial Numerators and Denominators A) Factor the numerator and denominator Divide out common factors Remember: Monomials can only be simplified with monomials and binomials can only be simplified with binomials that are exactly the same. B)
Rational Expressions When multiplying rational expressions, factor all the numerators and denominators. Divide out common factors. Factor all the numerators and denominators!
Rational Expressions When dividing rational expressions… 1)Keep, Change, Flip 2)Factor all the numerators and denominators 3)Divide out common factors Keep the first fraction Change division to multiplication Flip the second fraction (reciprocal)
Rational Expressions 1)When adding and subtracting rational expressions, find a common denominator if necessary. 2)Create equivalent fractions using the common denominator(Multiply by FOOs). 3)Add or subtract numerators and keep the denominator the same. 4)Simplify your final answer if possible. FORM OF ONE Ex:
Rational Expressions What is the sum of and expressed in simplest form? Add numerators and keep the denominator. Simplify by factoring the numerator and denominator.
Rational Expressions LCD ( Least Common Denominator ): 9x 2 Find equivalent fractions with a common denominator by multiply by a FOO (form of one). FOO
Rational Expressions 8x(x + 12) = 2x(2x + 4) 8x 2 + 96x = 4x 2 + 8x 4x 2 + 88x = 0 4x(x + 11) = 0 4x = 0 x + 11 = 0 x = 0 x = -11 FOO When solving rational equations (equations with algebraic fractions), combine fractions and set up a proportion. Remember: A common denominator is needed to add or subtract fractions. Reject 0 because it makes the equation undefined. Solution: x = -11 FOO
Scientific Notation If this a skill you have not mastered and need additional instruction, re-watch FLIP #3. Representing Numbers in Scientific Notation
Scientific Notation Multiplying Numbers in Scientific Notation Use the commutative property and laws of exponents Calculator Corner: 1)Press MODE 2)Select SCI (see top row), ENTER 3)Press 2 nd MODE to return to home screen 4)Enter expression into the calculator (use the expression on this slide to practice) (4x10^5)(3.2x10^-9) 5) Press ENTER 6) The expression 1.28E-3 means 1.28 x 10 -3 7)Go to MODE and select NORMAL to exit scientific notation
Scientific Notation Dividing Numbers in Scientific Notation Calculator Corner: 1)Press MODE 2)Select SCI (see top row), ENTER 3)Press 2 nd MODE to return to home screen 4)Enter expression into the calculator (use the expression on this slide to practice) 5)Put the numerator and denominator in ( ) (5.6x10^-12)/(4x10^6) 6) Press ENTER 7) The expression 1.4E-18 means 1.4 x 10 -18 8)Go to MODE and select NORMAL to exit scientific notation
Trigonometry Trigonometric Ratios What ratio represents the sine of the indicated angle pictured to the right? Answer: (1)
Trigonometry Finding Sides of a Right Triangle Use the Pythagorean Theorem when given two sides Use Trigonometry when given a side and an angle a 2 + b 2 = c 2 3 2 + 4 2 = c 2 9 + 16 = c 2 25 = c 2 5 = c Calculator must be in degrees (See MODE) Substitute known values into the trig ratio. Solve for the variable by cross multiplying. Pythagorean Theorem SOH CAH TOA
Trigonometry Finding Angles of a Right Triangle > Use inverse trig ratios Find the measure of the indicated angle to the nearest degree. Calculator: 2 nd TAN
Trigonometry Remember: Calculator must be in degrees in order to do a trigonometry problem. Go to Mode and highlight degree (ENTER) A 50 ft. ladder leans against a building. The foot of the ladder is 35 feet from the building. To the nearest degree, find the measure of the angle that the ladder makes with the ground. Are you looking for an angle or side? Trig Ratio or Inverse Trig Ratio? Draw a picture of the situation Looking for this angle
Now it’s your turn to review on your own! Use the information presented today to help you practice questions from the Regents Exams in the Green Book. See halgebra.org for the answer keys. Integrated Algebra Regents Review #2 Monday, June 16 th BE THERE!