Determine how many roots f (x) = ~4x5 + 6x' 13x3 5x2 12x + 102 have020313Explain your thinking:...

Determine which of the given numbers are roots of the given polynomial. $$1,1 / 2,2,-1 / 2,1 / 3 ; \quad f(x)=6 x^{2}+x-1$$

If X minus two is a factor of X cubed minus six X square plus nine x minus two, and we can find all the roots of this function if we first divide. Confirm that X minus two is a factor by showing that we get a remainder of zero. So we start the process X cubed, divided by access X Square district. You get executed minus two X where subtract excuse. Cancel negative six minutes. Negatives. Native six plus two or negative four X squared Bring down the nine X then we have negative four X distribute. We get negative four X squared plus eight x so tracked Cancels nine minus eight is one X minus two plus one and distribute the one we get X minus two. Subtract and cancel. Cancel it remainder of zero. So we know we have X minus two and expert minus four X plus one. So we know 20 But this is not fact herbal, but we can use the quadratic for me to find the roots exactly if a is one B is negative for and CS one so opposite be plus or minus the square root B squared, minus war a C all over to a or four plus or minus. This is 16 minus four, which is 12/2. That's four plus or minus. 12 is four times three sort of forest to to screw. Three is a square to 12 and this is the same thing. Is two plus or minus the square root of three or over to his two to over two was one Okay, What does this mean? But the roots are the original one hopes to because X minus two was a factor and then two plus the square root of three in two, minus a spirit of three.

Were given a function and were asked to find the zeros of this function. Algebraic Lee. The function is F of X equals the square root of three X plus two. Notice that our function is a radical function. Find the domain of dysfunction. You want to find the set of all real numbers X such that the arguments of the radical three X plus two is greater than or equal to zero. Solving this inequality for X. We get that This is the set of all real numbers such that X is greater than or equal to negative two thirds in an imitation. This is left bracket negative, two thirds positive infinity Now to find the zeros of dysfunction. Do you want to find a solution to the TV? Really evolved squares of three acts. What to This is a radical equation. Yeah, so he wants to get all the radicals on one side. He he already has. The next step is to square both sides, and you get zero equals three X plus two. They said they call it when you're dealing with radical equations in the square, both sides. You have to be careful of experience solutions now calling for X. We have that X equals negative, he said. And now we'll plug dispatch into our phasing to check for extremist solutions. We have that the square root of three times negative two thirds one who, if you do square roots of negative to prostitute with zero because the left side of the occasion therefore, this is not an extraneous solution. Also notice that negative two thirds just barely lies in our domain and therefore it follows that X equals negative. Two thirds is the one and only zero of our function Notice their function here is a radical function and not a polynomial. Therefore, the fundamental fear, um of algebra for Paulino meals defined on real numbers is not valid here.

Solution for the given question, we have to find out zeroes off the function. So solution. So all the solution. So look at the question they give the affects. They give the A perfect how to find out what might be the question. How to find out if the function is given how to find out this is of the function. First up was make a phone. Fax is equal to zero to find the zeros of the function. Whatever my bigger question first make, the function is equal to zero, then find the X value. Now, if if X is equal to cut the cushion, look at the question. It is X plus three. How old is invited by to Excess Square minus six. This is the question. Record the caution. Once again, prospects is equal. Do they have given express dream? Divided by holder two X squared minus six. Now make this function is equal to zero. That means X plus three divided about two extra square money. Six is equal to zero. So I make this function as they go to zero X plus three. They were red wine to X square, minus six is equal to zero from here. Find out X Well, that implies. What did you get him? Denominator Two X squared minus six and transpose that zero off. Zero in triple X squared 26 will become entire zero. So we express trees equal to zero and get expressed. Resicore 20 Therefore excessive corporal Ministry. So zeros of the function. This ministry, the fine lines arised ministry.

So we're looking at six x squared. Remind us nine X minus 60. So if we want the factors, I would factor out a three first that I loved. Toe two X squared, Uh, minus three X minus 20. You can check. My work is correct. By distributing that back in and then you can I doubt this is factor whole based off the directions. So I assume this is not fact herbal and use a quadratic formula now and understand that a is to B is negative three. And see his name of Tony, uh, can then figure out actually equal negative B plus or minus the square root of B squared. It's positive. Nine minus four times, eight times negative. 20. Um, so four times two is eight times to pay 16 with that negative in there. So plus 160. All over two A. Okay, so this was fact herbal. Um, because if you look at inside of that radical the square root of 169 uh, which we know the squares on your in 69 is 13 eso. The two answers we're looking for are three plus 13/4. Uh, which there you post 13 16 by my force for And then the other answer is three minus 13/4. So native 10 force or negative five halves. Um so these air the values that X could equal next could equal thinking if I have, but you want to be careful as you write as, ah, linear factor. So X minus forward, equal zero. What? This one some students think you can is right. Five halves on this. But you actually want to write this as ah two X plus five. And the reason like if you multiply everything by two there a reason before that, don't forget about the three in front. Um, X minus four into expose five. Um, is because you want to make sure that as you distribute this out, going name this f of X, Just don't leave room for myself in the front. Um, you want to make sure that as you foil this out, you get back to the original problem. Um, but let me circle their roots as well. There you go.