Find a polynomial function ƒ(x) of least degree with real coef...

Question

Find a polynomial function ƒ(x) of least degree with real coefficients having zeros as given. -2+√5, -2-√5, -2, 1

Accepted Answer

Hello, today we're gonna be solving for the polynomial that contains all of the given zeros. So the zeros that are given to us is negative three plus the square root of six, negative three minus the square root of six, negative five and four. Now keep in mind that these are all zeros of the polynomial that we want to solve for. So what this means is that the variable X is equal to all of the given zeros. That means that X is equal to negative three plus the square root of six, X is equal to negative three minus the square root of six, X is equal to negative five and X is equal to four. Now, in order to solve for the polynomial, we need to represent each of these equations as a factor of X. And in order to do that, we can set each of these equations equal to zero. On the left hand side, we can go ahead and add three and subtract the square of six to both sides of the equation. And doing this is going to give us X plus three minus the square root of six is equal to zero. In the next equation, we can add three and add the squared of six to both sides of the equation. And that is going to give us X plus three plus the square root of six is equal to zero. In the next equation, we can go ahead and add five to both sides of the equation to give us X plus five is equal to zero. And in the final equation, we can go ahead and subtract four to both sides to give us X minus four is equal to zero. Now, what we can go ahead and do is we can take the product of all the factors of eggs we just solved for. So if we go ahead and combine these together into a product, we get the quantity X plus three minus the square root of six times X plus three plus the square root of six times X plus five times X minus four. And if we take the product of all these factors of X, that is going to give us the original S square, the original polynomial that contains all the given zeros. So let's go ahead and foil this piece by piece. Let's foil the first two groups together and foil the last two groups together. Now, for the last two groups, we're going to be foiling together, X plus five and X minus four, we're going to distribute X the first term of the first group to each term in the second group X times X is going to give us X squared and X times negative four is going to give us negative four X. Then we're going to distribute the second term five to each term in the second group, five times X is going to give us five X and five times negative four is going to give us negative 20. Now, if we combine the like terms which is negative four X and positive five X, we can reduce this to a trinomial which is X squared plus X minus 20. So that's going to be the fold form of the last two groups. Then we're gonna repeat this process with the first two groups. So we're going to distribute the first term in the first group X to each term in the second group. So X times X is going to give us X squared X times three is going to give us three X and X times the square root of six is going to give us positive X square root of six. Then we're gonna repeat this process for the second term in the first group which is three and distribute it to each term in the second group. Three times X is going to give us three, X, three times three is going to give us nine and three times. The square root of six is going to give us three square root of six. And finally, we're going to distribute the negative square of six to each of the terms in the second group. So negative square root of six times X is going to give us negative X times the square root of six, negative square root of six times three is going to give us negative three square root of six and negative square root of six times square root of six is going to simplify to just give us negative six. Now, what we want to go ahead and do is combine the light terms together X squared of six minus X squared of six is going to cancel each other out. Three, X plus three, X is going to give us six X and three squared of six minus three square to six is going to cancel each other out. And finally, we can go ahead and add the constants which are nine and negative six to give us positive three. So that means that the first two groups are gonna foil together to give us X squared plus six X minus six. So if we rewrite the next group with this first term, the first polynomial is going to simplify to give us X squared plus six, X minus six times X squared plus X minus 20. Now, what we're going to do is we're going to foil this group together and that is going to give us the polynomial that we are looking for. So we're going to distribute X squared to each term in the second group X squared times X squared is going to give us X to the fourth X squared times X is going to give us X cubed and negative 20 times X squared is going to give us negative X squared. Then we're going to distribute the second term to each term in the second group. So six X times X squared is going to give us six X cubed, six X times X is going to give us six X squared and six X times negative 20 is going to give us negative 120 X. Then we're going to distribute the last term which is negative six to each term in the second group. Negative six times X squared is going to give us negative six X squared, negative six times X is going to give us negative six X and negative six times negative 20 is going to give us positive 120. Now, all we need to do is we need to go ahead and add the like terms together. Adding the light terms together is going to give us a simplified polynomial which is in the form of X to the fourth plus seven X cubed minus 11 X squared minus 100 and 17 X minus 60. And this is going to be the original polynomial that contains all of the given zeros. And with that being said, the answer to this problem is going to be a So I hope this video helps you in understanding how to approach this problem. And I'll go ahead and see you all in the next video.

Find a polynomial function ƒ(x) of least degree with real coefficients having zeros as given. -2+√5, -2-√5, -2, 1

Key Concepts

Polynomial Zeros and Factors

Complex Conjugate Root Theorem

Constructing the Polynomial of Least Degree