Factor the following polynomial and determine the domain, range, and x-intercepts: y=x^2-3x+2
To factor the polynomial, guess and check is used. The terms have to start with an x term because there is a x^2 in the polynomial. Because there is a positive for the last term an a negative for the x term, we know there will be two negative signs multiplied together to create the last positive term. So far, we know (x-_)(x-_)=x^2-3x+2. In order to result in +2, the blank terms need to be 1 and 2. If we multiply out (x-1)(x-2) to check our work, it results in x^2-x-2x+2=x^2-3x+2. Once the polynomial is factored, the x-intercepts occur when y=0. So, set the factored polynomial equal to 0. (x-1)(x-2)=0. In order to equal 0, one of the two terms must equal 0. Therefore, x must equal 1 or 2 for y=0. x=1 and x=2 are the roots of the function. The domain of the function is the values of x that result in a valid y output. Because there is no possibility of the imaginary number because of a square root, the domain is all real numbers. You can also determine the domain by graphing the function. The range of the function is the y values and it is limited by the function and the values of x. You can determine the range by plugging in various values of x into the polynomial. If x=-1, y=6; if x=0, y=2; if x=1, y=0; if x=1.5, y=-0.25; if x=2 y=0; if x=3, y=2, etc. As shown by the values, the minimum value of y is -0.25 and the maximum is infinity. Therefore the range is from -0.25 to infinity. You can also determine the range by graphing the function.
Simplify the two fractions listed below, convert them to decimals and percentages and then find the Least Common Multiple (LCM) of the fractions. 1. 12/24 2. 15/20
To simplify the fractions, find a common number that the denominator and numerator can be divided by. For problem 1, the common number is 12 so the fraction reduces to 1/2. For problem 2, the common number is 5 so the fraction reduces to 3/4. The fractions convert to 0.5 and 0.75. The percent is a 100*Decimal value which results in 50% and 75%. The least common multiple of the reduced fractions is when a multiple of one fraction equals the multiple of the other fraction. The first multiple to overlap between fractions is the least common multiple. Multiples of 1/2 are 2/2, 3/2, 4/2=2, 5/2, 6/2=3 etc. Multiples of 3/4 are 6/4=3/2, 9/4, 12/4=6/2=3. The first multiple that both fractions have is 3/2, so that is the LCM.
Using the order of operations, evaluate the following equations: 1. (6*3-12/4)/5 2. sqrt(-19+7*5)
1. Step one is to simplify the numbers within the parenthesis because of the order of operations (also known as PEMDAS: Parentheses, exponents, multiplication, division, addition, subtraction). Within the parentheses, the multiplication and division operations are completed before the subtraction according to PEMDAS. Once simplified it equals (18-3)/5. Step two is to simplify even further for 15/5 which then equals 3. 2. Step one is to follow the order of operations and simplify within the parentheses. Within the parentheses, the multiplication operation is completed first according to PEMDAS. This results in sqrt(-19+35). Step two is to complete the addition of the numbers inside the parentheses for sqrt(16) which equals 4.