Find a polynomial function ƒ(x) of least degree having only real ...

Question

Find a polynomial function ƒ(x) of least degree having only real coefficients and zeros as given. Assume multiplicity 1 unless otherwise stated. See Examples 4–6. 2-i and 6-3i

Accepted Answer

Hello, everyone using the zeros, 10 minus I and eight minus four. I, we've been asked to write the equation of a polynomial function of the least degree. The coefficient of the polynomial must be only real and take multiplicity one. Unless otherwise stated, we are given four polynomials as our answer choices to begin. I'm going to write my F of X function. So I get F of X equals and I'm gonna put an A for a coefficient that we might be able to find. We might not, but we need a placeholder for. And then I'm gonna write each factor based on the zeros we know. So open parentheses, my first factor is X minus the first zero we're given, which is 10 minus I. So in parentheses, 10 minus I close two sets of parentheses. Since the zeros we're given are complex numbers. We need to use what is called the conjugate root theorem, which means that if a complex number is a root, its conjugate is also a root. So in this case, this means we have a factor that is also X minus 10 plus I as that's the conjugate of 10 minus I our next factor is again gonna be X minus and the zero we were given. So eight minus four, I closed parentheses and we have one more zero or one more factor because we have the conjugate of eight minus four. I, so this factor will be X minus eight plus four. I, so right now we have F of X equals a multiplied by the factor X minus 10 minus, I multiplied by the factor X minus 10 plus I multiplied by the factor X minus eight minus four. I multiplied by the factor X minus eight plus four. I, I'm gonna go through and distribute the negative sign into the complex numbers for each set of parentheses. So F of X equals a then open parentheses. I have X minus 10 and now it's plus I closed parentheses. Next open parentheses, X minus 10 minus, I close parentheses. Next, open parentheses, X minus eight plus four. I close parentheses and finally open parentheses, X minus eight minus four. I closed parentheses. We have a lot of multiplication to do here. I'm going to multiply two trinomial at a time. So I'm gonna begin by multiplying X minus 10 plus I by X minus 10 minus I. When you do this, make sure you distribute each term from the first trinomial into the second trinomial and then be cautious to combine all of your like terms. And when you do so you get, so I have F of X equals A and it's gonna be times the product of those two parentheses, which is X squared minus 20 X plus 100 minus I squared. Next, I'm multiplying together X minus eight plus four, I and X minus eight minus four. I again, be very cautious to distribute each term from the first trinomial into the second and then combine all of your like terms. And when you do that, we should end up with X squared minus 16, X plus 64 minus 16, I squared closed parentheses. So at this step, I again have F of X equals a multiplied by open parentheses. X squared minus 20 X plus 100 minus I squared, closed parentheses multiplied by open parentheses. X squared plus 16, X plus 64 minus 16, I squared. One simplification step that I left to be done was I squared. Recall that I squared is the same as negative one. So in our first set of parentheses where we have minus I squared, this is like saying plus one. So I'm gonna rewrite those parentheses and I'm gonna add the one to the 100. So I have a multiplied by the parentheses. X squared minus 20 X plus 101 closed parentheses. And in our second set of parentheses, we have negative 16 I squared. So again, if I squared is the same as negative one, we can rewrite negative 16 I squared as plus 16. So now we're gonna combine those like terms in my second set of parentheses, I have X squared minus 16 X and now 64 plus 16 gives me 80 one more set of multiplication. To do. I need to multiply together X squared minus 20 X +10 101 by X squared minus 16 X plus 80. So again, please be very cautious when you multiply this in, make sure you distribute each term from the first trinomial into the second and combine your like terms. And when you do so, our equation now will read F of X equals a multiplied by X raises to the fourth power minus 36 X cubed plus 501 X squared minus 3216 X plus 8080 closed parentheses. Now, to address the A, we don't know any other values. So we're going to assume a is one. So this makes our final function F of X equals X to the fourth power minus 36 X cubed plus 501 X squared minus 3216 X plus 8080. That is our polynomial that goes with those zeros right here and it matches check our signs very closely. Um It matches answer choice. D, we can be very careful because all of them have slightly different signs. But it is answer choice. D have a nice day.

Struggling with College Algebra?

Find a polynomial function ƒ(x) of least degree having only real coefficients and zeros as given. Assume multiplicity 1 unless otherwise stated. See Examples 4–6. 2-i and 6-3i

Zeros of Polynomial Functions practice set