* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download Worksheet
Survey
Document related concepts
List of important publications in mathematics wikipedia , lookup
Wiles's proof of Fermat's Last Theorem wikipedia , lookup
Brouwer fixed-point theorem wikipedia , lookup
Four color theorem wikipedia , lookup
Fermat's Last Theorem wikipedia , lookup
Fundamental theorem of calculus wikipedia , lookup
Factorization of polynomials over finite fields wikipedia , lookup
Proofs of Fermat's little theorem wikipedia , lookup
Vincent's theorem wikipedia , lookup
Transcript
Rational Root Theorem Name: Period Date: How do we find the factors of a polynomial? 1. Factor the following quadratics (a) p( x) x 2 x 30 (b) p ( x) 4 x 2 5 x 6 2. How did you decide which numbers to try as factors? Did you use the Factor Theorem? How would you extend your thinking to factor higher order polynomials? 3. Given the polynomial f ( x) x 4 3x3 15x 2 19 x 30 . If we were to factor this polynomial as ( x a)( x b)( x c)( x d ) , what would be the possible values of a, b, c and d? Try to find the values of a, b, c, and d by using the Factor Theorem and dividing out each factor as you successfully find it. Graph the function on your calculator to check your answers. 4. When the zeros are all integers it is easy to use the calculator, but if they are not all integers it is hard to tell if it factors or if the solutions are irrational, (i.e. one of the factors is a quadratic that can’t be reduced further using only integer coefficients) The polynomial equation 3x 5 x 26 x 8 0 has a solution which is not an integer, but it is a rational number. To use your calculator to help you find the solutions in such a case, set the xscl to 1/3. (Why would we do this?) Set the window to: Xmin = -5 Ymin = -20 Xmax = 5 Ymax = 60 Xscl = 1/3 Yscl = 1 3 2 If we look at the graph we see that it crosses the x-axis one notch to the right at x twelve notches to the left at x 4 . 1 , six notches to the right at x 2 , 3 Write this equation in factored form and prove the factoring works. 5. Use a similar technique to solve 2 x x 31x 15 0 (what is your x-step here?) 3 2 D:\840968319.doc 6. If x p is a zero of a polynomial then (qx-p) is a factor of the polynomial. Why is this? q Fill in the blanks: The Rational Roots Theorem: Given a polynomial f(x) the only possible rational solutions of the equation f(x) = 0 are x p . Where p is a factor q of…………………….…… and q is a factor of …………………………… 7. Use the rational roots theorem and the factor theorem to factor the following polynomials (you may use your calculator as much as you like). (a) x4-3x2-4x+12 (b) 2x4-5x3-14x2+5x+12 The Remainder Theorem. If a polynomial p(x) is divided by a linear binomial ( x c) the remainder will always be p(c). 8. (a) 9. Perform the following divisions and then evaluate p(c) in each case to satisfy yourself of the truth of this theorem: p( x) x3 x 2 x 5 divided by ( x 1) 134 What would the remainder be if 2 x (b) p(x)=3x3-12x2-9x+1 divided by (x-5) 7 x 45 x9 4 is divided by ( x 1) ? D:\840968319.doc