To assist us in finding an integer zero of the polynomial we use the following result. One of the main methods of solving quadratic equations was the method of factoring. Similarly, one of the main applications of factoring polynomials is to solve polynomial equations. Since the product of the three factors is zero, we can equate each factor to zero to find the solutions.
Thus the number of distinct solutions may be less than the degree, but it can never e x ceed the degree. In some situations, the factorisation results in a quadratic equation with either no real solutions or irrational solutions. In this case, we may need to complete the problem by using the quadratic formula. Note that there are polynomial equations with irrational roots that cannot be solved using the procedure above.
In general, factoring polynomials over the integers is a difficult problem. In the module, Quadratic Functions we saw how to sketch the graph of a quadratic by locating. The verte x is an e x ample of a turning point. For polynomials of degree greater than 2, finding turning points is not an elementary procedure and usually requires the use of calculus, however:.
To get a picture of the overall shape of the curve, we can substitute some test points. We can represent the sign of y using a sign diagram:. It does not tell us the ma x imum and minimum values of y between the zeroes.
Notice that if x is a large positive number, then p x is also large and positive. If x is a large negative number, then p x is also a large negative number. If we e x amine, for e x ample, the size of x 4 for various values of x , we notice. In the case of the parabola, we call this a verte x but we do not generally use this word for polynomials of higher degree. Instead we talk of a turning point and further classify it as a ma x imum or minimum.
In the following we will consider odd powers greater or equal to 3. As above, the graph is flat near the origin. At the origin we have neither a ma x imum nor a minimum. The sign diagram is. The zeroes of a polynomial are also called the roots of the corresponding polynomial equation. To properly understand how many solutions a polynomial equation may have, we need to introduce the comple x numbers.
The comple x number i is often referred to as an imaginary number. Every polynomial equation of degree greater than 0, has at least one comple x solution. Every polynomial equation of degree n , greater than 0, has e x actly n solutions, counting multiplicity, over the comple x numbers.
E x plain how the corollary may be deduced from the theorem. Hence, every polynomial of degree n , greater than 0, can be factored into n linear factors using comple x numbers. However, the equation has only two distinct roots. The verte x of a parabola is an e x ample of a turning point. The x -coordinates of the turning points of a polynomial are not so easy to find and require the use of differential calculus which is studied in senior mathematics.
We can perform a similar e x ercise on monic cubics. These identities give relationships between the roots of a polynomial and its coefficients.
The study of equations of degree greater than two goes back to Arabic mathematics. It was not until the Renaissance that the general solution of the cubic was obtained. We now e x pand the left-hand side and factor 3uv from two of the terms to give. At this stage, we have two numbers u 3 , v 3 whose sum and product we know. Use your calculator to e x press this in decimal form and check that it satisfies the original equation. He discovered a method to reduce the problem of solving a quartic to that of solving a cubic.
In both cases it is possible to e x press the solution of the given equation using square and higher roots and the usual operations of arithmetic addition, subtraction, multiplication and division. Such a solution is often called a solution using radicals.
In the 18 th and 19 th centuries, the great mathematicians, Euler, Lagrange, Eisenstein and Gauss further e x tended our understanding of polynomials and polynomial equations. This led to the development of what is nowdays called modern algebra which is concerned with the study of algebraic structures. In particular, suppose p x is a polynomial with degree greater than 0, and real coefficients,. The fundamental theorem of algebra is used to show the first of these statements.
To obtain the second, we need to know the fact that when we have a polynomial with real coefficients, any comple x roots will occur in pairs, known as conjugate pairs. This fact can be used to prove the second statement.
This requires a little knowledge of comple x numbers. Eisenstein c. Suppose that we can find a prime number p that does not divide the leading coefficient an, but which does divide all of the other coefficients.
That is, p x is irreducible. E x plain how to construct a polynomial of arbitrarily large degree that cannot be factored over the rationals. These are called power series.
Thus, for e x ample,. The notation n! Thus 5! These infinite series are often referred to as Maclaurin series and have very wide application in both mathematics and physics. There still remain today unsolved problems related to polynomials. The appendi x below discusses in broad outline a remarkable application of polynomials to modern telecommunications. An application of polynomials to error-correcting codes. When your mobile phone sends or receives messages, or data is sent from satellites deep in space, information may be lost or corrupted along the way to its destination.
This code can detect one error, but cannot correct it. A polynomial modulo 2 is a polynomial whose coefficients are either 0 or 1. This polynomial cannot be factored modulo 2 since the only possible roots are 0 and 1 and neither work. We now suppose that the polynomial m 1 t has a root a. We use these as the coefficients of the polynomial. Take the message 1, 0, 0, 1 and encode it as 1, 0, 0, 1, x , y , z.
Converting to a polynomial, we have. Hence we encode the message 1, 0, 0, 1 as 1, 0, 0, 1, 1, 1, 0. Suppose that one error occurs in the fifth number from the left so we receive the message 1, 0, 1 , 1, 1, 1, 0. Thus the t 4 coefficient is incorrect and to the polynomial for the received message is.
In general, if there were e x actly one error in the th digit, we would receive. This process, of course, assumes that at most 1 error occurred. Assuming at most one error, correct and decode the message 1, 0, 0, 1, 0, 0, 1. Although the order of the terms in the polynomial function is not important for performing operations, we typically arrange the terms in descending order of power, or in general form.
The degree of the polynomial is the highest power of the variable that occurs in the polynomial; it is the power of the first variable if the function is in general form.
The leading term is the term containing the highest power of the variable, or the term with the highest degree. The leading coefficient is the coefficient of the leading term. We often rearrange polynomials so that the powers are descending. When a polynomial is written in this way, we say that it is in general form. Identify the degree, leading term, and leading coefficient of the following polynomial functions.
The leading coefficient is the coefficient of that term, —4. The leading coefficient is the coefficient of that term, 5. Knowing the degree of a polynomial function is useful in helping us predict its end behavior. To determine its end behavior, look at the leading term of the polynomial function. Because the power of the leading term is the highest, that term will grow significantly faster than the other terms as x gets very large or very small, so its behavior will dominate the graph.
For any polynomial, the end behavior of the polynomial will match the end behavior of the term of highest degree. Describe the end behavior and determine a possible degree of the polynomial function in Figure 7.
We can describe the end behavior symbolically by writing. In words, we could say that as x values approach infinity, the function values approach infinity, and as x values approach negative infinity, the function values approach negative infinity. We can tell this graph has the shape of an odd degree power function that has not been reflected, so the degree of the polynomial creating this graph must be odd and the leading coefficient must be positive.
Describe the end behavior, and determine a possible degree of the polynomial function in Figure 9. The degree is even 4 and the leading coefficient is negative —3 , so the end behavior is. In particular, we are interested in locations where graph behavior changes. A turning point is a point at which the function values change from increasing to decreasing or decreasing to increasing.
We are also interested in the intercepts. As with all functions, the y- intercept is the point at which the graph intersects the vertical axis. The point corresponds to the coordinate pair in which the input value is zero. The x- intercepts occur at the input values that correspond to an output value of zero. It is possible to have more than one x- intercept. A turning point of a graph is a point at which the graph changes direction from increasing to decreasing or decreasing to increasing.
The y- intercept is the point at which the function has an input value of zero. The x -intercepts are the points at which the output value is zero. The y- intercept occurs when the input is zero so substitute 0 for x. The x -intercepts occur when the output is zero. The y- intercept occurs when the input is zero. To determine when the output is zero, we will need to factor the polynomial. We can see these intercepts on the graph of the function shown in Figure The degree of a polynomial function helps us to determine the number of x -intercepts and the number of turning points.
A polynomial function of n th degree is the product of n factors, so it will have at most n roots or zeros, or x -intercepts. The graph of the polynomial function of degree n must have at most n — 1 turning points. This means the graph has at most one fewer turning point than the degree of the polynomial or one fewer than the number of factors.
0コメント