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Write a polynomial equation with integer coefficients that has the given roots

Quadratic equations with complex solutions Video transcript We're asked to solve 2x squared plus 5 is equal to 6x. And so we have a quadratic equation here. But just to put it into a form that we're more familiar with, let's try to put it into standard form.

Fundamental theorem of algebra Video transcript Voiceover: So we have a polynomial right over here. We have a function p x defined by this polynomial. It's clearly a 7th degree polynomial, and what I want to do is think about, what are the possible number of real roots for this polynomial right over here.

So what are the possible number of real roots? For example, could you have 9 real roots? And so I encourage you to pause this video and think about, what are all the possible number of real roots? So I'm assuming you've given a go at it, so the Fundamental Theorem of Algebra tells us that we are definitely going to have 7 roots some of which, could be actually real.

So we're definitely not going to have 8 or 9 or 10 real roots, at most we're going to have 7 real roots, so possible number of real roots, so possible - let me write this down - possible number of real roots. Well 7 is a possibility. If you graphed this out, it could potentially intersect the x-axis 7 times.

Now, would it be possible to have 6 real roots? Is 6 real roots a possibility? Is this a possibility? Well, let's think about what that would imply about the non-real complex roots.

If you have 6 real, actually let's do it this way. Let me write it this way. So real roots and then non-real, complex. The reason I'm not just saying complex is because real numbers are a subset of complex numbers, but this is being clear that you're talking about complex numbers that are not real.

So you could have 7 real roots, and then you would have no non-real roots, so this is absolutely possible. Now could you have 6 real roots, in which case that would imply that you have 1 non-real root. Well no, you can't have this because the non-real complex roots come in pairs, conjugate pairs, so you're always going to have an even number of non-real complex roots.

So you can't just have 1, so let's rule that out. Now what about having 5 real roots? That means that you would have 2 non-real complex, adding up to 7, and that of course is possible because now you have a pair here.

So I think you're starting to see a pattern. Essentially you can have an odd number of real roots up to and including 7. So for example,this is possible and I could just keep going. I could have, let's see, 4 and 3. This is not possible because I have an odd number here.

You're going to have to have an even number of non-real complex roots. So rule that out, but then if we go to 3 and 4, this is absolutely possible. You have two pairs of non-real complex roots.

And then we can go to 2 and 5, once again this is an odd number, these come in pairs, so this is impossible. And then you could go to 1 real and 6 non-real. Completely possible, this is an even number.

And then finally, we could consider having 0 real and 7 non-real complex and that's not possible because these are always going to come in pairs, so you're always going to have an even number here.

So the possible number of real roots, you could have 7 real roots, 5 real roots, 3 real roots or 1 real root for this 7th degree polynomial.We will start with the closed-form formulas for roots of polynomials of degree up to four.

For Equation (1) has three roots: y1 = A+B, An exact test was given in by Sturm, who showed how to count the real roots within any given range of values.

Edit Article How to Factor a Cubic Polynomial. Two Parts: Factoring By Grouping Factoring Using the Free Term Community Q&A This is an article about how to factorize a 3 rd degree polynomial. We will explore how to factor using grouping as well as using the factors of the free term.

Write a factored form polynomial functionf(x) of least degree that has a leading coefficient of 1 with the real zeros shown in the graph. WITHOUT a calculator, sketch the graph of each polynomial function using the information provided.

The Content - It's not just about batteries. Scroll down and see what treasures you can discover. Background. We think of a battery today as a source of portable power, but it is no exaggeration to say that the battery is one of the most important inventions in the history of mankind.

A Time-line for the History of Mathematics (Many of the early dates are approximates) This work is under constant revision, so come back later. Please report any errors to me at [email protected] In Exercises write a polynomial equation with integer coefficients that has the given roots.

Y= -2, y = 3, y = 3 (3 is a double root.).

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Polynomials: The Rule of Signs