Kasten, Algebra 2 Finding Zeros (Roots) of Quadratics, Cubics, and Quartics A zero of a polynomial equation is the value of the independent variable (typically x) that, when plugged-in to the equation, makes the equation zero. For example, consider the following quadratic equation: y = x 2 1 If you plug in 1 for x (in other words, let x = 1) then y = ( 1) 2 1 = 1 1 = 0. So, we say that x = 1 is a zero of the equation y = x 2 1. Finding the zeros of a polynomial equation can be challenging! Usually, we are given nice quadratic equations that we can factor easily using strategies we learned in Algebra I. To find all of the zeros in our previous example, we can simply factor the quadratic equation: Now, let y = 0: 0 = (x + 1)(x 1) y = x 2 1 = (x + 1)(x 1) To solve for x, we will use the zero product property, which says that if the product ab = 0, then either of the factors, a or b, must be 0. For our example, we have (x + 1)(x 1) = 0. By the zero product property this means that either (x + 1) = 0 or (x 1) = 0. Solve for x in each case: and, x + 1 = 0 x = 1 x 1 = 0 x = 1 Thus, the zeros of y = x 2 1 are x = 1 and x = 1.
Finding Zeros without Factoring For more complicated equations (especially equations that aren t quadratic), it s often impossible or difficult to factor. In these cases, our calculators make life a whole lot simpler. To find zeros of equations that can t be factored, simply graph the equation on your TI-84 (or similar calculator). I ve graphed y = x 2 1 for this example. Once the equation is graphed, we can use a builtin calculator program to find the zeros by pressing the following keys: 2ND: TRACE:
Then, select zero from the menu. The calculator then asks you to set a left bound. To do this, simply move the point left on the graph until the point is to the left of your x-intercept. Once your cursor is to the left of the x-intercept, press enter. The calculator will then ask you to set a right bound. To do this, move your cursor to the right of the x- intercept. Then, press enter.
Finally, the calculator will ask you to guess where the x-intercept is. To do this, move your cursor as close as you can to the x-intercept and press enter. Note: your cursor must be between the left and right bounds that you set earlier. If the three steps are followed correctly, the calculator will show you the zero at the bottom of the screen. So, we can see that one of the zeros of y = x 2 1 is x = 1. To find the other zero, repeat the process but for the x-intercept on the right-hand side. You should get something close to x = 1. IMPORTANT: Sometimes, we don t get nice, round answers using this technique. Depending on the equation or the window settings of your graph, you may get only an approximate answer. Round appropriately.
Practice: finding zeros using factoring or your calculator 1. y = x 2 + 2x + 1 2. y = x 3 6x 2 + 1 3. y = 8x 4 3x 2 (You may need to resize your window or zoom for this graph)
Rational Expressions A rational equation is an equation that features rational expressions. Rational expressions are fractions. We ve dealt with rational expressions since Algebra I, but it s often helpful to review the basics so that we have a strong foundation from which to work. Rational expressions pop up constantly in higher-level math classes, so it s important to master the basics! Reducing Just as we can reduce simple fractions (for example: 2 4 can be reduced to 1 ), so too can we reduce 2 fractions that involve variables. We call such fractions rational expressions, and we use basic fraction rules to add, subtract, or perform any mathematical operation on them. To reduce a fraction, we must factor the top, factor the bottom, and then cancel the common factors. For example: 10a + 15 8a 2 + 10a 3 = 5(2a + 3) (2a + 3)(4a 1) = 5 4a 1 The big trick in all of this is to keep in mind that we can never, ever, ever, ever divide by zero. So, we have to note the values of a that will make the denominator zero. To do this, simply take the original denominator, set it equal to zero, and solve for a: By the zero product property, we have: 8a 2 + 10a 3 = (2a + 3)(4a 1) = 0 and 2a + 3 = 0 2a = 3 a = 3 2 4a 1 = 0 4a = 1 a = 1 4 Thus, in order to never divide by zero, a 3 2 or 1 4
Let s try another example: 4 b Reduce b 2 16 Solution: 4 b b 2 16 = 4 b (b 4)(b+4) = 1(b 4) = 1 (b 4)(b+4) b+4 Finding the excluded values of b, set the denominator equal to zero and solve for b: (b 4)(b + 4) = 0 b 4 or 4 Thus, we reduce the fraction to 1, b 4 or 4 b+4
Practice: reduce the rational expressions 1. 4c 6 d 7 e 8 6c 2 d 7 e 10 2. 2x 2 x 2 1
Adding and Subtracting Rational Expressions The rules for adding or subtracting rational expressions are the same as they are for adding or subtracting fractions: Fractions must have the same denominator before we can combine them Combine the denominators (make one fraction with the same common denominator) and add or subtract only the numerators Example: Add 1 2 + 1 3 Solution: 1 2 and 1 do not have a common denominator. 3 To find a common denominator, we find the least common multiple (LCM) of both denominators, 2 and 3. The LCM of 2 and 3 is 6. In other words, 6 is the smallest number we can find for which 2 and 3 are both a factor. Now, we can t just change the denominators of each fraction willy-nilly; we have to use a trick that will give us the same fractions, 1 2 and 1, but with the common denominator of 6. 3 To accomplish this, we multiply each fraction by a special one. Multiply 1 2 by 3 3 to get 3 6. Note that 3 = 1. We haven t really changed the fraction, we ve just multiplied by a particular 6 2 expression of one, 3, to change the appearance of the fraction. 3 Do the same to 1 3, multiplying the fraction by 2 2 to get 2 6. We now have 3 6 + 2 6 Combining the denominators and adding the numerators, we get 3 + 2 6 = 5 6
Now, let s look at an example where we have variables and parentheses thrown into the mix: Example: Kasten, Algebra 2 Subtract 1 a+1 a b Solution: Since we have two different denominators, a and b, we need to get a common denominator before combining the fractions. To do this, we find the least common multiple (LCM) of a and b. This is simple enough; we just multiply a and b to get ab. As in the previous example, multiple each fraction by a special one : We get: ( b b ) (1 a ) (a + 1 b ) (a a ) b (a + 1)a ba ba We can then combine the fractions and simplify: b (a + 1)a ba = b a2 a ba Notice how the subtraction sign worked: We distributed the subtraction sign through the numerator of the second fraction! It s just as if we multiplied the second fraction by 1. Always take care when subtracting fractions to distribute the negative through the numerator. Also note: Sometimes, we are given fractions that have a common denominator from the start. If this is the case, simply combine them into a single fraction with the common denominator and add or subtract the numerator.
Practice: add and subtract rational expressions 1. 4x+5y 7 + 2x 7y 7 2. 2a 3b 6a 4a 2b 9b (find the LCM of 6a and 9b before combining) 3. 7 6ab 3 + 5 8a 2 b