Projectile Motion Example: After the semester is over, Herman discovers that the math department has changed textbooks (again) so the bookstore won't buy back his nearly-new book. Herman goes to the roof of the math building, which is 160 feet high, and chucks his book straight down at 48 feet per second. How many seconds does it take his book to strike the ground? Use the formula h(t) = –16t 2 – 48t + 160 I need to find the time for the book to reach a height of zero ("zero" being "ground level"), so: 0 = –16t 2 – 48t + 160 Factor out -16 so it is in standard form. 0 = -16(t 2 + 3t – 10) You can now divide both sides by -16 to simplify. t 2 + 3t – 10 = 0 (t + 5)(t – 2) = 0 t = –5 or t = 2 t = -5 does not make sense because time cant be negative, so Correct answer is t=2 seconds. CONCLUSION: It takes 2 seconds for the book to strike the ground.
CHAPTER 8 RATIONAL EXPRESSION REVIEW A rational expression is a fraction in which the numerator or denominator is a variable expression (such as a polynomial). A rational expression is undefined if the denominator has a value of 0. A rational expression is in SIMPLEST form when the numerator and denominator have no common factors other than 1. Reducing to simplest form – factor the numerator and denominator, then cancel out any common factors in the numerator and denominator (not common factors that are both in the numerator or both in the denominator, e.g. side by side). Multiplying Rational Expressions – factor the numerators and denominators then cancel out common factors as above, then multiply the numerators and multiply the denominators. Dividing Rational Expressions – change to a multiplication problem by changing the DIVISOR into its RECIPROCAL. Example: Put each polynomial in standard form, flip divisor (expression to the right of ÷) then completely factor each polynomial After factoring, cancel out common FACTORS between numerators and denominators
Adding and Subtracting Rational Expressions – Step 1: Factor the denominators, then find the LCM. The LCM of two polynomials is the simplest polynomial that contains the factors of each polynomial. To find the LCM of two or more polynomials, first factor each polynomial completely. The LCM is the product of each factor the greater number of times it occurs in any one factorization. Step 2: Change each rational expression so that the new denominator will be the LCM. You will multiply the numerator and denominator of each expression by whatever it takes to get the LCM as the new denominator. Step 3: Add the two new fractions by adding the numerators and keeping the denominator (the LCM) the same. Step 4: Now factor the resulting expression and cancel out any common factors in the numerator and denominator. Simplify Complex Fractions – Complex fractions are just rational expressions with fractions within fractions. To simplify, find the LCM of all the denominators of every fraction in the expression, then multiply the main numerator and denominator by that LCM. Then simplify as usual. LCD= x 3
WORK Rate of Work * Time Worked = Part of Tasked Completed If someone can do a job in 60min, their rate of work is 1/60min. If someone else can do the same job in 40minutes, their rate of work is 1/40min. The TIME to get the same job done TOGETHER can be found by Adding their parts together to make 1 whole job. Solving Equations with Fractions – multiply BOTH SIDES of the equation by the LCM of all denominators in the equation. Then solve as usual. Make sure you know the restrictions. The solution cannot be a value that is not in the domain. That is, it cannot be a value that would make any of the original fractions undefined. If the equation is one fraction set equal to another, this is called a PROPORTION. Solve by CROSS-MULTIPLYING, then isolating the variable. x = 0 or x = -1 would make one of these fractions undefined. Cross-multiply Use dist. prop. to simplify. If equation is degree 2, then it is a quadratic equation. Put it in standard form (0 on one side). Factor and use zero-product rule to solve for x.