VIEWS: 5 PAGES: 4 POSTED ON: 11/16/2011 Public Domain
Algebra II CP1 7.5 Zeros of Polynomial Functions A polynomial is a FACTOR of another polynomial if __________________________________ ________________________________________________________________________ Example: Use substitution or synthetic division to determine if x + 1 is a factor of P( x) 3x3 4 x 2 x 5 Yesterday we found some roots using the calculator and long division. We will do the same today, with some tougher answers Example: Find all of the rational roots of 10 x3 9 x 2 19 x 6 0 . Examine the graph of the related function P( x) 10 x 3 9 x 2 19 x 6 on your graphing calculator and test all solutions using synthetic division or long division. One root of P( x) 0 appears to be -2. Test whether P(2) 0 is true. From the graph of the related function, there appears to be two real zeros or one real zero with a multiplicity of 2 between 0 and 1. (“multiplicity of 2” refers to a factored situation like ( x 3)( x 3) where 3 is a solution twice) A closer look between 0 and 1 on the graph of the related function shows 2 zeros…find them on the calc and test them using synthetic division. 2nd possible root= 3rd possible root= Test using synthetic division: Factored Equation: ______________________________________ ALL the zeros are ______, ______, _______ *If a function is a cubic (power of 3) then it has at least one real zero. If it is a quartic (power of 4) then it has at least 2 real zeros. Use this as your starting point when asked to find all zeros of a polynomial function. You can often use the quadratic formula to solve a polynomial equation. Example 2. Find all the zeros of Q( x) x 3 2 x 2 2 x 1 Sometimes you will find that your equation only crosses the x-axis once, when you originally thought it would cross three times (since it is a cubic). When this happens, it means that two of your roots are IMAGINARY!! Example 3. Find all the zeros of P( x) 3x 3 10 x 2 10 x 4 Now that you know how to find the roots, let’s work backwards. Let’s say that I GIVE you one or two of the roots, and ask you to expand it to find the original polynomial. Example 4: Find the Q(x) if two of the roots of the equation are -2 and 4 + i. First, find the OTHER complex root: _____________ Now, write the equation in FACTORED form: Q(x) = _________________________ Time for some distribution! I would distribute the two complex roots first, then distribute the real root. Be careful with your signs while multiplying! Q(x) = Example 5. Find all the zeros of P( x) x 3 6 x 2 7 x 2 Example 6. Find all the zeros of P( x) x 3 9 x 2 49 x 145 Homework: page 464 #’s 23-33 odd (check back of the book for correctness…but show all work!)