Continuing our discussion of analytic geometry and trigonometry, in this segment, I want to talk about polynomials and conics. So, first of all, a polynomial equation is an equation that contains variables which we sometimes call indeterminates and coefficients. And for a single determinant or variable, let's say, x, the general form of the equation is an, x to the n, plus an to the minus one, x to the n minus one, etc. And in a polynomial equation, the powers or the indices, are integer numbers which are positive. And the coefficients a and an minus one etc are constants. In other words, they're numbers. In particular, we deal often with the quadratic equation which has this form, ax squared plus bx plus c is equal to zero. And the roots or the solutions to this equation are given by x is equal to minus b plus or minus, square root of b squared minus 4ac divided by 2a, are the roots or the solutions to that equation. And there are two of them. And furthermore, there are two real and unequal roots if the quantity b squared minus 4ac is positive, in other words greater than zero. And two other useful relationships that don't appear in the handbook are the some of the roots x1 plus x2 is equal to minus b over a. And the product of the roots x1 times x2 is equal to c over a. So, let's do an example on that. The roots of the quadratic equation minus 9x plus x squared is equal to minus 18 are which of these pairs? First, we put it into the standard form, which is ax squared plus bx plus c is equal to zero. And the equation becomes x squared minus 9x plus 18 is equal to zero. From which we see that the values of the coefficients are a is equal to one, b is minus nine, and c is equal to 18. And the roots of the equation from our general solution are, minus b plus minus etc. And substituting in the values of a, b and c into that equation, we get this from which we see, that the roots are six and three, and the answer is c. We also note that we can factor this equation in this way, now that we know the roots x minus six times x minus three is equal to zero. So, for that equation to be zero, either x minus six or x minus three is equal to zero, from which it's obvious that x is equal to either six or three, which is the answer that we already obtained. And I suggest that you try expanding this out and show that, indeed, you'd get back to the original equation right here. Next, the handbook discusses quadric surfaces. In other words, spheres, and more generally, this refers to quadratic surfaces such as spheres, ellipses, etc. But apparently, in the handbook, we only discuss spheres. So, the equation of the surface of the sphere which is centered at coordinates h, k and m at radius r is given by this expression right here. And an example on that, the radius of the sphere centered on the origin that passes through the point with coordinates two, three, and four, is most nearly which of these? So, starting with the general equation or the standard form of the equation for the sphere, we have r squared is x minus h squared, etc. But in this case, we're given that the sphere is centered on the origin. Therefore, h, k, and m are all zero. So, substituting in the coordinates here, we have x is equal to two, y is equal to three, z is equal to four, so r squared is equal to 29, so half the radius is the square root of that, or 29. So, the answer is C. Next, we'll look at conics. And conics are plane surfaces which are formed by passing a plane through a cone. And this is the sketch which is obtained from the reference handbook showing the different sorts of sections that you can find and the general equation of a conic. Here is a better diagram, which illustrates them a little bit better. And shows the four different types of surfaces or conics that you can form. So, firstly, if the plane through the cone is horizontal, in other words, parallel to the base, we get a a circle. If the plane is tilted at some angle but does not pass through the base of the cone, it's an ellipse. If the surfaces are tilted and passes through the base of the cone to the left of the central point, it's a parabola. And finally, if the surface is tilted, passes through the base to the right of the center, we form a hyperbola. And the general conditions for these to occur is given in the reference handbook as shown here. So, here is the general form, again, which I've reproduced, where both a and c are not zero. So, which type of surface you get depends on the value of b squared minus 4ac. If b squared minus 4ac is negative, less than zero, it's an ellipse. If it's positive, it's a hyperbola. If it's zero, it's a parabola. And if a is equal to c and b is zero, it's a circle. And finally, if a, b and c are zero, it's a straight line. This is the general form of the normal form of a conic section which has a principal axis parallel to a coordinate axis, the x or y axis. So, in this case, we have a more general form of the equation and there are four different cases here which are given in the handbook in terms of the so called eccentricity. E which is defined here as cosine theta over cosine FE. So, the first case, a parabola has eccentricity E equals one, and here is the general equation of a parabola. Case two, an ellipse has eccentricity less than one, it's this case here. And the form of the equation then is given by that. Case number three, the hyperbola has E eccentricity greater than one and here is the equation for the hyperbola. And the last case, the circle. Eccentricity E is equal to zero, in other words, a horizontal plane is a circle. And the general equation of a circle is given by this equation right here. So, let's do an example on that. The conic section described by this equation 2x squared minus 2y squared plus 6x plus 3y equals ten, is which of those forms, circle, ellipse, a hyperbola or a parabola? So, in this case then, here's our equation of the conic. And if we compare that with the most general form, Ax squared plus Bxy plus Cy square etc, is equal to zero. By comparing those two equations, we see that the values of the coefficients are as follows. A is equal to 2, and B is equal to zero, etc. So, computing the values, first of all, we see that the coefficients A and C are not zero and the value of B squared minus four AC is given by this expression here, which is equal to 16 which is positive, it's greater than zero. Therefore, we conclude that the shape of this conic is a hyperbola. Hyperbola, which is this case right here, it looks like that. And this concludes our discussion of polynomials and conics.
