Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Name ______________________ Susan E. Wagner High School Mr. Gary Giordano, Principal Ms. L. Silver, AP Department of Mathematics Mr. R. Gargano, Teacher MR21 – Review Notes for Exam 1-2MP Solving problems using the Least Common Denominator: To find the LCD … factor denominators to obtain prime factors; then take all the different prime factors and raise them to the power which is equal to the highest number of times the factor appears in any ONE denominator. Uses… I: Addition / Subtraction (keep the LCD as the new denominator) Arithmetic Example 1. Prime factor the 2 denominators (10 and Add 12). Prime factors of 10 are 5 • 2. Prime factors of 12 are 3•2•2 Find the LCD 2. There are 3 different prime factors: 5, 3, and 2. The most 5 and 3 appear in any one denominator is once; but the most 2 appears is twice. Therefore the LCD will be 5•3•2•2, which equals 60. Raise the numerator by the fraction’s new factors. Thus = and = + Algebraic Example 1. Prime factor the 2 denominators. Prime factor a2 – 1 and get (a – 1)*(a + 1). The prime factor for a + 1, is itself. Add Find the LCD = = 2. There are two different factors, (a – 1) and (a + 1). They appear at most only one time in any denominator. Therefore, the LCD is (a – 1)*(a + 1) which equals a2 – 1 Raise the numerator by the fraction’s new factors Thus, and = = So, + = = OR II. To simplify a complex fraction (a fraction which has a fraction as the numerator and/or denominator. 1. Simplify 2. Multiply the numerator and the denominator of the complex fraction by the LCD. This is to eliminate the denominators in the complex fraction. 3 and 2 are the prime factors so the LCD = 2(3) = 6 (I put the 6 over 1 just to see this example more clearly: of course, ) Simplify Multiply the numerator and the denominator of the complex fraction by the LCD. This is to eliminate the denominators in the complex fraction. The prime factors of the “small” denominators are “2” and “x”; so the LCD = 2x 3. Multiply by the LCD to remove the small denominators and then cancel. 2 3 = (answer… and note that the original denominators, the “2” and “3” were removed), so the numerator and the denominator are whole numbers.) The reduced fraction is the answer. III. To solve equations which contain fractions. The easiest way to solve fractional equations is to remove the denominator. Do this by multiplying each term in the equation by the LCD. 1. Determine the LCD for the fractions… Solve for the LCD is 2. Multiply each term by the LCD and the denominators will be canceled. 3. After canceling multiply across each numerator with the factor(s) that were not canceled. This will give an equivalent equation without denominators. Then solve this equation. 4. Because of the fact that fractional operations are not valid if denominators are 0, we must check the result. So the solution is