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Finding the Zeros of a Polynomial Function 1

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we're going to find all the zeros of the function f of x equals x to the fourth plus 5x cubed + 4x squared + 20 x so if this equation Y is equal to 0 like instead of being a function with f of x if it was an equation like x to the fourth plus 5x cubed + 4x squared + 20 x equals e we would do the same process so if we are asked to find the zeros the function or if we're asked to find the roots of an equation we will use the same either way so the first thing you want to do is use your graphing calculator and type the polynomial under y equals into your graphing calculator you can take a look at the table or you can take a look at the grass but if you look at the table what you're looking for is the numbers where there's a y value of 0 so you should see in your graphing calculator if you type this in and it would be a good idea right now to just type this in your graphing calculator see what you see and try to work it out on your own and then if you have a hard time or you get stuck go ahead and play the video and watch how to work it out and if you're not stuck work all the way to the end fast forward and check your answer and then if you got it right move on and if you got it wrong go back and watch I put this into my graphing calculator I get zeros at -5 and 0 so what we're going to try to do it shrink this problem down it's right now we're starting out to the fourth power so we want to take that degree down two squared we want to take it down to a quadratic so what I want to do is use these two zeros that my calculator gave me with synthetic division and shrink the problem down so I'm going to start with so when I use synthetic division on put -5 on the outside and you going to put your coefficients make sure that you have a complete countdown 4 3 2 1 we're missing a constant so we're going to have to put a zero at the end for our constant so we have a 1 of 5 of 4 a 20 and then we need to put a 02 and for the set which is missing from the function bring down your one and multiply -5 * 1 is -5 and you get 0 and then -5 times 0 is 0 4 + 0 is 4 -5 * 4 is -20 that gives us a 0 and then -5 times 0 is 0 0 + 0 is 0 so that's kind of weird how it came out but we started with x 4th and now we're down to exit third so this would be our cubed turn the first term and then this would be our quadratic term and are linear term and our continent is so so so now we are trying to does to be shrunk down to x squared so I'm going to do the same thing again I'm going to get rid of these exes but I'm going to use synthetic division again with the other zero that we found in our table I already found the x equals negative 5 that's going to be one of our Solutions so now we're going to do the same with zero now don't go back to the beginning and start back with your original coefficients do it with this. best polynomial that you've gotten when you are shrunk it down to exit the third okay so bring your one this is going to be weird cuz you're multiplying by 0 but 0 x 1 is 0 + 0 is 0 0 x 0 is 0 4 + 0 is 4 * 400020 so we got a good read because you got a remainder of zero and that's what we were expecting because we had a zero in our table there so now so we started with a fourth power at the top and then we divided and I brought it down to X cubed and now we're at a square so we want to write this expression out as a polynomial so this is our tax and this is our constant the fours a constant x squared plus for anyone to solve this to remember we're finding the zeros or where the Y value is zero or if the equation is equal to 0 so we're going to set it equal to zero to solve can't be factored even though it's tempting because they're both perfect squares this is not a subtraction sign so that's a deal-breaker we can only Factor difference of squares not sum of squares so we can't Factor so we're just going to use the square root property this also we're going to subtract 4 so that gives us a squared equals -4 then we're going to take the square root of both sides don't forget to put a plus or minus when you take the square root so we have x equals Placer - that's going to represent two solutions one positive one negative and we're going to take the square root of 4 which is two and then we're going to have an eye for the square root of -1 so we have those to salute and we have x equals negative 5 and then we also have x equals 0 so we expected to get for Solutions because the degree of the polynomial or and we did our Solutions are negative 5 0 2 I + -2 I or you could write that as negative 5 0 + r - 2

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Finding the Zeros of a Polynomial Function 1

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