a-third-degree-polynomial-function-f-has-real-zeros-2-frac12-and-3-and-its-leading-coefficient-is-ne

Name: a third degree polynomial function f has real zeros 2 frac12 and 3 and its leading coefficient is ne
Uploaded: 2019-03-28
Duration: 1 min 2 s
Channel: Numerade
Description: A third-degree polynomial function $f$ has real zeros $-2, \frac{1}{2},$ and $3,$ and its leading coefficient is negative. Write an equation for $f .$ Sketch the graph of $f .$ How many different polynomial functions are possible for $f ?$

Question

A third-degree polynomial function $f$ has real zeros $-2, \frac{1}{2},$ and $3,$ and its leading coefficient is negative. Write an equation for $f .$ Sketch the graph of $f .$ How many different polynomial functions are possible for $f ?$

Frank Lin · Accepted Answer

So this problem are one twenty nine. We want the start different polynomial with real zero Annette it to one of our two three. So let&#x27;s try this. Don&#x27;t do explosive to practice X minus one over two months X minus story but Russell on the reading coefficient to be naked. So that&#x27;s also for the negative one from the That&#x27;s a question. But what? Have and, uh, the profits like this come in different polynomial function, or possibly even a different vision off again, preventing a number here and there will be a different story, says other conditions.

Heather Zimmers · Answer

in this problem. We&#x27;re going to put together these pieces of information and sketch a graph and write in the equation paragraph. It needs to be CUBIC. So its third degree, it has a negative leading coefficient, and that indicates the end behavior is going to be the left and down on the right end up. And he has euros at negative 21 happened three. So we can start by plotting those points negative to 1/2 and three. Okay, let me go back to the end behavior. I have that upside down. I gave you the end behavior for positive leading coefficient. It would be more like that for the end behavior. So we have something like this. Okay, now let&#x27;s write in the equation for that. So first of all, let&#x27;s start with their zeros. X equals negative, too. X equals 1/2 and X equals three. So let&#x27;s go ahead and change these equations so that they all say equals zero. So we add to to both sides and we get X Plus two equals zero. We multiply both sides by two and subtract one and we get two x minus. One equals zero, and we subtract three from both sides and we get X minus three equals zero. This is how we find our linear factors. So now we can multiply those linear factors together. X plus two times two X minus one tens X minus three If we want to multiply it all the way out, we&#x27;re probably gonna start by boiling on two of them. So that would give us two X squared plus three X minus two. And then we&#x27;re gonna multiply that by X minus three being systematic about it. I&#x27;m going to start with two x squared and multiply it by both terms in the second a polynomial. So I get two x cubed minus six x squared. And then I&#x27;m going to take the second term and multiply it Bible. And that gives me three X squared minus nine x. And then I&#x27;m going to take the third term and multiply by both. And that gives me minus two X plus six. And then we can combine the like terms. So we have two x cubed minus three X squared, minus 11 x plus six. And that&#x27;s a function. However, it doesn&#x27;t have a leading coefficient, so it&#x27;s good to have a leading coefficient. We need to multiply it by negative one or by negative anything. So we can say that our function f of X is negative. Two x cubed Last three x squared plus 11 x minus six. Now that&#x27;s just one possibility. We could multiply that by any real number, and we&#x27;ll just get a vertical stretch or shrink of this function so it would have the same zeros as this one. So I should say we shouldn&#x27;t We could multiply that by any positive real numbers. Remember, we need to maintain the negative leading coefficient and eso They&#x27;re infinitely many possibilities.

Ankit Gupta · Answer

it is given there. How many negative riel rules? Negative. Really. Zeros are rules. Our route are possible for a polynomial f. So everything polynomial and then given that f minus X hedge radiation five times. So that means this is the negative number off. Negative rules rules. This is maximum decision five and then less by even number that three or one. So these are the what&#x27;s will number off this deal, the possible number off negative rewards. It will defy it. Maybe three, maybe one.

Suman Saurav Thakur · Answer

we have about another 50. Given that Zito&#x27;s as one minus 10 and F two equal to three. Let us right F X equal to a X minus, one X plus one and x. Okay, this is one. Now, if affect after equal to tree, so stable people toe a will plug in to in place of X two miles. One is one de and do even be called through three by six. That is one by two. So let us plug in one Affects will be equal to one by two X x minus one x plus one. Thank you so much.

Ankit Gupta · Answer

here we have given the degree over function effects as speak was too full and three zeros as three place do I and Ford and four. Now, as we all know, that complex groups comes with its consecrate sold. Another route will be three miners who are now writing the function. F X is X minus full into X minus food, which is X minus four Squire and X minus three minus. Do I and X minus three plus two I now relating this function F X is X minus. Full Squire X minus. See Squire plus four. So this is the final function ethics and the unsettled this question.

Suman Saurav Thakur · Answer

we have a number 52 individuated to find polymer function effect and it we have given it zeroes. Zeros has been given toe minus three and five and also after people music. So let the poor normal function affects is a x minus two x plus three x minutes. Fire No, we have F three after he will be the value but plugging in X equal to 33 minus two three plus three minus five. We have after equal to six equal to a into one into six into minus two, so it will be equal to six by minus 12. That is minus one by two equal to minus on by two so portable function will be affects. Equal to minus one by two x minus two X plus three and x minus Fight. Thank you.

Frank Lin · Answer

in this problem. We&#x27;re going to put together these pieces of information and sketch a graph and write in the equation paragraph. It needs to be CUBIC. So its third degree, it has a negative leading coefficient, and that indicates the end behavior is going to be the left and down on the right end up. And he has euros at negative 21 happened three. So we can start by plotting those points negative to 1/2 and three. Okay, let me go back to the end behavior. I have that upside down. I gave you the end behavior for positive leading coefficient. It would be more like that for the end behavior. So we have something like this. Okay, now let&#x27;s write in the equation for that. So first of all, let&#x27;s start with their zeros. X equals negative, too. X equals 1/2 and X equals three. So let&#x27;s go ahead and change these equations so that they all say equals zero. So we add to to both sides and we get X Plus two equals zero. We multiply both sides by two and subtract one and we get two x minus. One equals zero, and we subtract three from both sides and we get X minus three equals zero. This is how we find our linear factors. So now we can multiply those linear factors together. X plus two times two X minus one tens X minus three If we want to multiply it all the way out, we&#x27;re probably gonna start by boiling on two of them. So that would give us two X squared plus three X minus two. And then we&#x27;re gonna multiply that by X minus three being systematic about it. I&#x27;m going to start with two x squared and multiply it by both terms in the second a polynomial. So I get two x cubed minus six x squared. And then I&#x27;m going to take the second term and multiply it Bible. And that gives me three X squared minus nine x. And then I&#x27;m going to take the third term and multiply by both. And that gives me minus two X plus six. And then we can combine the like terms. So we have two x cubed minus three X squared, minus 11 x plus six. And that&#x27;s a function. However, it doesn&#x27;t have a leading coefficient, so it&#x27;s good to have a leading coefficient. We need to multiply it by negative one or by negative anything. So we can say that our function f of X is negative. Two x cubed Last three x squared plus 11 x minus six. Now that&#x27;s just one possibility. We could multiply that by any real number, and we&#x27;ll just get a vertical stretch or shrink of this function so it would have the same zeros as this one. So I should say we shouldn&#x27;t We could multiply that by any positive real numbers. Remember, we need to maintain the negative leading coefficient and eso They&#x27;re infinitely many possibilities.

Problem

Use a graphing utility to graph the function give…

Problem 129 Easy Difficulty

A third-degree polynomial function $f$ has real zeros $-2, \frac{1}{2},$ and $3,$ and its leading coefficient is negative. Write an equation for $f .$ Sketch the graph of $f .$ How many different polynomial functions are possible for $f ?$

Answer

$-2 x^{3}+3 x^{2}+11 x-6$
Infinitely many
See graph

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$-2 x^{3}+3 x^{2}+11 x-6$Infinitely manySee graph

Precalculus with Limits

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Alayna H.

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Michael J.

$-2 x^{3}+3 x^{2}+11 x-6$
Infinitely many
See graph

Ron Larson, David C. Falvo

Precalculus with Limits