This document provides a 3-step process for finding all the zeros of a function: 1) Factor the function using synthetic division. For the example function F(x)=X3-6x2+11x-6, this yields F(x)= 1x2-5x+6. 2) Factor the resulting quadratic function using the quadratic formula or completing the square. For the example, this gives the zeros as x-3 and x-2. 3) Identify the zeros by writing the factored function and graphing it. For the example, the zeros are 2, 2+i, and 2-i.

1. Finding all of the zeros of a functionBy: Ashley Ezell

2. Step OneFactor the function with synthetic divisionF(x)=X3-6x2+11x-61_| 1 -6 11 -6 1 -5 6 1 -5 6 0F(x)= 1x2-5x+6

3. Step TwoDo a diamond/box problem to factor this functionDiamond of 6x2 at the top and -5x on the bottom is -3x and -2xWhen you plug these numbers in the box you get x-3 and x-2, set these equal to zero and they become positive

4. Step Two cont.An easier way to do this step is to use the quadratic formula-Only if function is set up as ax2+bx+cEx. X2-4x+5 Quadratic formula: X= -b±√(b)2-4(a)(c) 2(a)-(-4) ±√(-4)2-4(1)(5) = 4 ± √-4 2(1) 2simplify: 2 ± i

5. Step 3Identify Zero’s 2 ± iF(x)= (x-2)(x-2+i)(x-2-i)↑ ↑ ↑ Opposite sign!GRAPH!!!