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### Course: Algebra 2 (Eureka Math/EngageNY)>Unit 1

Lesson 17: Topic D: Lessons 38-40: Complex roots and the fundamental theorem of algebra

# Quadratics & the Fundamental Theorem of Algebra

The proof of the Fundamental Theorem of Algebra for any degree of polynomial is really tough. For now, let's note that it indeed holds for polynomials of the second degree (i.e. quadratics). Created by Sal Khan.

## Want to join the conversation?

• If a real root crosses the x axis, a complex root crosses...? What does it cross? And, can it be seen?
• No, the complex root is not defined by crossing anything, and no, on our regular x,y graph it can't be seen. But if we make a vertical axis of the units of i (imaginary numbers), and make the real numbers run along a horizontal axis, the complex root can be seen graphed on this "complex plane." (And all sorts of fun can be had with graphing on the complex plane in precalculus)

Also, even though the complex roots don't exist on the x,y graph (since there's no such thing as a square root of a negative real number, so there's no way we can plot it on the cartesian plane of real numbers, that's why we need the imaginary plane) it can still be visualized on the x,y graph using a little trick, which is, to take our parabola and flip it upside down. This reflected parabola now has to cross the x=axis, and have 2 real roots.
If this flipped parabola has the equation y= -(x-a)^2+b, then it has the real roots a +/- sqrt(b) whereas our original parabola had the equation y- (x-a)^2+b with the complex roots a +/- sqrt(b)*i.
So, since on a complex plane its the vertical axis that has the imaginary numbers, to visualize it on our normal graph, we take the roots of this flipped parabola, make a perfect circle with the points of the two roots on the opposite sides of this circle, then rotate the points 90 degrees, as shown on this website: http://www.math.hmc.edu/funfacts/ffiles/10005.1.shtml
Hopefully this last bit wasn't confusing, it's just a way to pretend we have the complex numbers on the x,y graph
• What about when y = x^2. Isn't there only one root(0).
• Yes, but only because +0 and -0 are not two different numbers. 0 is unique in that it is its own opposite.
• Assuming you have two complex roots and a real root, how do you get the function from those roots?
• If the roots of a polynomial are at x = a, x = b and x = c (whether a, b and c are real or complex) then an equation with those roots will be (x - a)(x - b)(x - c) = 0. Then you can multiply out the brackets if you wish. Beware, though. This may not be the original function, because you can also multiply the whole thing by ANY non-zero number (real or complex) and the resulting polynomial will still have the same roots.
• At , couldn't he just simplify and write
1/5(i) instead of -3/5 + 4/5 (i)
• In the expression: -3/5 + 4/5 (i), the 2 terms are unlike terms. We can only combine like terms. Since the first term has no "i", it is "unlike" the 2nd term.
If the expression had been: -3/5(i) + 4/5 (i), then you could combine them into 1/5 (i)
Hope this helps.
• is it possible for a quadratic equation to not touch either the x or y axis? How would you solve this?
• it's not possible. all quadratic equations have a y intercept at their c value, although some will not touch the x axis.
• What does the complex root that is found mean? For example, the root that was found was x = -(3/5) + -(4/5)i . What does that tell us about the graph or equation?
(1 vote)
• There isn't anything too important about nonreal complex roots that shows on the standard 2-D graph because we are not plotting the nonreal complex values of x, but only the real complex values. However, if you graphed the function in the complex plan, with the imaginary portion being one of three axes, then you would see the place where the f(x) or y value of the function reaches 0.

As far was what it might mean in a real-world application, there are numerous applications of nonreal complex numbers, especially in electrical engineering, weather forecasting, and even in computer science.
• But it may only have one real root, if it touches the x-axis once like y=(x-1)^2 would
• In this case, there are two roots, but they are both equal to each other. y = (x - 1)(x - 1). We call this a root with a multiplicity of 2, since it appears twice in the function's factored form. Roots with multiplicity are counted however many times they need to be for the fundamental theorem of algebra. You can see if a root is like this (even multiplicity) visually if the line of the function "bounces back" from the x-axis after touching it, like how the parabola approaches the x-axis as you get to the vertex, and then it goes back in the opposite direction. Roots with a multiplicity of 1 will just go straight through. If you have a root with an odd multiplicity that isn't 1, such as in y = x^3, the graph will lessen its curve before it goes through the x-axis and resumes its behavior on the other side.
• what happens when the discriminant equals 0 for 2nd degree polynomials ?
• When the discriminant = 0, then the quadratic equation has one root (or x-intercept) that is a real number.
• This is a question from the course challenge (precalculus):
z^3 = -64
It states that one root is (-4) and asks us to find the other 2 complex roots.