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Transcript
Advanced Functions 3.1 Remainder and Factor Theorems Name______________________ Date __________ We’ve learned how to multiply polynomials. Now, we’ll learn how to DIVIDE polynomials. There are 2 ways to divide: long division and synthetic division. First, let’s go back to grade school and review how to divide plain old regular numbers. 8 65180 Now, let’s apply the same process to polynomials. x - 2 x 2x 6x 9 3 Try 2 2x + 3 2 x 7 x 17 x 3 3 2 Check your solution by multiplying the divisor by the quotient and adding the remainder. Synthetic division works differently, but is pretty cool! 3 2 Divide x 2 x 6 x 9 by x – 2 1. First, the divisor must be in the form: (x – k) So, in our example, k = 2. What is k in the divisor (x + 4)? 2. You only need the coefficients of the terms. However, if a term is missing, you must use a zero as a place holder. 3. Write the leading coefficient. Then multiply and add, multiply and add, etc. 2 1 2 2 -6 8 -9 4 1 4 2 -5 4. The answer is interpreted as follows: Work backwards. The last number is the remainder. The next number back is the constant. The next number back is the x coefficient. 2 The next number back is the x coefficient. 3 The next number back is the x coefficient. And so on. So, the answer above is 2 1 1 x Try it. 2 2 2 4 -6 8 2 -9 4 -5 x c remainder Divide x 3 2 x + 4x + 2 + 14 x 8 by x + 4 5 x2 Remainder Theorem: In the example on the previous page, divide x 2 x 6 x 9 by x – 2, we got x + 4x + 2 + 3 2 2 5 x2 The Remainder Theorem states that f(2) will be equal to –5! Try it. Factor Theorem: The factor theorem just uses definitions you already know! Solutions = Roots = Zeros = X-Intercepts When we find the roots of a polynomial equation, we are finding the places where the value of the function is ZERO. If f(x) = 0, then x is a root!!!! Now we have to connect this idea with synthetic substitution. Remember that synthetic substitution allows us to find the value of a function for a given value of x. Watch what happens when we find f(3) for the function f(x) = x2 + 2x − 15 3 1 2 −15 3 15 1 5 0 The value of the function is 0!! That means 3 is a root of the equation, a zero of the function, and an x-intercept on the graph!! That also means that (x − 3) is a factor of the polynomial. And it means that (x + 5) is also a factor, also know as a reduced polynomial. Wow!! Getting zero in synthetic substitution is a big deal!! Study this example: f(x) = x3 + x2 + 2x + 24. Graph the function. (You will need to set your window.) There is one real root at −3. Because the degree of the polynomial is 3, we know there are two other roots. They must be imaginary. We will use synthetic substitution to divide out (x + 3). Then the quadratic formula will allow us to find the other roots. −3 1 1 2 −3 6 1 −2 8 24 −24 0 So, using the coefficients of the quotient, we write (x + 3)(x2 − 2x + 8) = 0 We already know x = − 3. We use the quadratic formula 2 4 4(1)(8) 2 28 on the second ( ) and get = = 2 2 2 2i 7 = 1 i 7 . So, x = −3, 1 i 7 , 1 i 7 2 This stuff is not hard. It just takes practice to remember the process! (AKA – Do your HW!)