How To Find Roots In A Polynomials

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1 Quadratic Functions We are reacquainted with quadratic equations, this time as a function. A quadratic function is of the form f ( x) = ax + bx + c, which is also called the general form. We know already that the graph of a quadratic function is a parabola, and we also already know that a quadratic function may have 0, 1, or Real solutions, depending on how many times the graph crosses the x-axis. We will make use of shifts, reflections and stretches. Quadratic functions can be rewritten in the form f ( x) = a( x h) + k, which allows us to make use of the vertical shift k, horizontal shift h and reflection/stretch factor a. This is called the standard form of a quadratic function. In this form, we can immediately read off important qualities about the graph: its vertex (h,k), whether it faces up (a > 0) or down (a < 0), and whether it is stretched ( a > 1 or a < 1) or shrunk ( 1 < a < 1). To rewrite a quadratic function from general to standard form, you must complete the square. Suppose a parabola has vertex (1,) and passes through (3, 6). We note the vertex is (1,), meaning that f ( x) = a( x 1) +. To solve for a, use the other known point on the graph ( 3, 6), insert for x and y, and solve for a to get a =. The final answer is f ( x) = ( x 1) +. Since a < 0, the graph is reflected, opening down. Since a < 1, it is stretched slightly in appearance. The vertex of a parabola is useful in applications since it is the maximum point if the graph opens down, or the minimum point if the graph opens up. Polynomial Functions of Higher Degree This section reviews general polynomial functions and their graphs. Please experiment by typing various polynomial functions into your calculator and viewing the graphs. We make some general observations: 1. All polynomial functions are smooth and continuous.. Polynomial functions of even degree always face up or face down, depending in the value of the leading coefficient (similar to a reflection). 3. All polynomials of odd degree start high and end low if the leading coefficient is negative, or start low and end high if the leading coefficient is positive. Last and very important: 4. A polynomial of degree n has at most n x-intercepts; hence it has at most n roots. (The words zero, x-intercept, root and solution are synonymous)

2 This last principle means that a parabola (degree ) has at most roots and never 3, for instance. A cubic (degree 3) has at most 3 roots but never 4. It is possible for the polynomial to have less than n roots, but never more. Keep in mind that determining roots for higher degree polynomial functions is sometimes very difficult. You can solve a linear equation simply, and the quadratic formula solves every quadratic equation, but there is no easy formula for a cubic or quartic (degree 4). Other methods must be used. There actually do exist formulas for solving the cubic (known as Cardan s Formula) and the quartic (known as Ferrari s Formula). These formulas are so big and cumbersome that they are essentially useless in practice and really are nothing more than curiosities. There do not exist general formulas to solve general quintics or higher. Girolamo Cardano, who derived the general solution to the cubic in the 16 th century, was a fascinating, bizarre man who is well worth a brief exploration on the internet. The Intermediate Value Theorem (IVT), is often used to show the existence of a root, but 3 not necessarily its value. For instance, let f ( x) = x + x 3. We note that if x = 1 then f ( 1) = 1 and if x = then f ( ) = 7. Since f (x) is continuous, and since it travels from the negative y s to the positive y s within the interval [1,], we must conclude that there exists a root in that interval. In other words, there exists some value c in [1,] such that f ( c) = 0. What is this c? The IVT does not tell us. But it does tell us conclusively that this mysterious c value does exist. The IVT is often used in Calculus. At this stage in the process, you ll be given higher-ordered polynomials that can be factored using the techniques we have already seen. But in general, not all polynomials can be factored easily. We will see how to attack them shortly. Real Zeros of Polynomial Functions Long Division of Polynomials is an algorithmic process exactly similar to regular long division of numbers. The point is to break up a large polynomial (the dividend) into two smaller factors (the divisor and quotient). A remainder is entirely possible. However, if the remainder is 0 the dividend has been successfully factored into polynomials of smaller degree. Determining solutions then is very easy. The Division Algorithm of Polynomials formally presents this concept. Synthetic Division is a shorthand method of polynomial long division, and simply requires practice in order to accomplish the process. But what if we are not given a k value initially? Do we just guess until we find one that works? There is a better method. The rational zero test allows us to create a candidate list of possible rational roots of f (x). At the very least, it narrows the field considerably. You still need to check each candidate value to see if it works. The rational zero test, sometimes called the method of p s and q s, works like this:

3 1. Write the polynomial f (x) in general form, in descending powers. Set p equal to the last coefficient, the constant, and set q equal to the leading coefficient.. Create a set of all possible ratios of the factors of p over the factors of q. Consider the negative versions of each value as well. 3. Start testing each value until you find one that works. For example, consider f ( x) = 4x + x x + 6. We set p = 6 and q = 4. The factors of p are {1,, 3, 6} and the factors of q are {1,, 4}. Therefore, the set of possible rational roots of f (x) is {±1, ± 1, ± 41, ±, ± 3, ± 3, ± 43, ±6}. You then start testing each one by synthetically dividing through f (x). Suggestion: If you successfully synthetically divide a polynomial, try that same value again on the new quotient! It just may work again. In this case you have what s called a repeated root. If f (x) is to have a rational root, it will be of the form p q, as in above. Otherwise f (x) f (x) to have an irrational root, which won t show has no rational root. It is possible for via this test, or no real roots at all. Question: A polynomial of odd degree must always have at least one real root. Can you explain why? Consider the types of graphs created by odd-degreed polynomials. Always augment your search for roots by graphing the function and visually finding roots. There are other methods and tricks to help determine roots to a polynomial. However, with the availability of graphing calculators, most of these methods have become obsolete. The Fundamental Theorem of Algebra (FTA) We are now ready for the coupe de gras: The Fundamental Theorem of Algebra. It is a simply stated theorem: Given a polynomial of degree n > 0 with real coefficients, we are guaranteed at least one root and at most n roots. The roots may be complex. This theorem is very profound: it guarantees the existence of a root for every polynomial. It doesn t tell how to find the roots, but that the roots exist. The FTA was proven by C. F. Gauss for his doctoral dissertation in the late 18 th century. It is not a long proof, but requires an advanced knowledge of mathematics to fully understand the steps. We are still left with the task of locating these roots. We have these options at hand:

4 1. Use the QF on all quadratics. This is where the complex roots may appear.. Use your grapher, the rational zero test, and synthetic division to break down larger polynomials. A corollary of the FTA is that all polynomials of degree n with real coefficients can be completely factored into linear factors. This shows a nice relationship between a polynomial and its roots. We are reminded that complex roots always come in conjugate pairs. If we know that a + bi is a root to some polynomial, then we can conclude that a bi is also a root. You can now rest easy knowing that all polynomials are at the very least solvable. Rational Functions and Asymptotes p( x) A rational function is a function of the form f ( x) = q( x) ; essentially it is a fraction of functions. Because a rational function is not a polynomial, the FTA does not guarantee the existence of roots for such equations. However, often the top and bottom functions are polynomials, in which case we can locate roots. We see an immediate concern: if the denominator is zero, then f (x) is undefined. Therefore, we must make the requirement for any rational function that x cannot be chosen such that the denominator equals 0. If the denominator is a polynomial, use any of the various methods you have learned to solve for 0, and then simply state that x can not equal these values. x 1 For example, if f ( x) = x +, we say that x. We are restricting the domain of in this manner. The domain of (x) is x x. f { } f (x) Note: When stating a domain like the previous example, avoid statements like all x except x. That s actually a double negative! Think about it. Sometimes the denominator has no Real roots, in which case the domain is not restricted. x 3 An example is g ( x) = x. If we set x + 1 = 0, we get only the two roots ± i. Since we + 1 don t normally plot complex numbers, we allow x to be any Real number. The domain for g (x) is { x x R}. (The fancy R is just the normal symbol for the Reals. You can also just say All x. If a rational function has a root, it occurs where the numerator equals 0, as long as the denominator is not also 0 at this point. When we solve the denominator for 0, we get certain values of x that can not be used. These values of x become vertical asymptotes.

5 Fact: A rational function never crosses a vertical asymptote. The horizontal asymptote is determined by the degrees of the numerator and denominator. It is best thought of as the long-term behavior of the graph; in other words, how does the graph behave as x gets large positively or negatively. Fact: A rational function with polynomial numerator and denominator has at most one horizontal asymptote, but possibly many vertical asymptotes. Fact: A rational function can cross a horizontal asymptote. Graphs of Rational Functions We will now discuss in depth the methods of graphing a rational function. With practice, you can graph these functions without a grapher and with minimal effort. We usually start by determining all vertical asymptotes and sketching these on a coordinate axis using dashed lines. We then determine any horizontal asymptotes and graph it as well. Next, we determine the x-intercepts by setting the numerator equal to 0 and solving for x. Determine the y-intercept by evaluating f (0). We now have some asymptotes partitioning the coordinate plane and have plotted the intercepts. At this time, plot a few points to determine the behavior of f (x) between each asymptote. Be warned: Your calculator sometimes gives unsatisfactory plots of rational functions (Usually losing a lot of the detail). Compare your grapher s plots versus your own handdrawn plots. The grapher is nice but it does have limitations. An interesting type of asymptote is known as the slant (or oblique) asymptote. This occurs when the degree of the numerator is exactly one more than the degree of the denominator. Long divide (or synthetically divide) the denominator into the numerator; the quotient will be the equation of the slant asymptote. Rational functions can have a wide assortment of appearances, as you may have noticed from all the pictures in this section.