Class 10 Notes I. Polynomials A. Defintion B. Standard form II. Solving and factoring polynomial equations III. Graphs A. Limits B. Zeros I. Polynomials A. Definition In this last week we will tackle a final kind of function, called a polynomial. A polynomial is a function f(x) that can be written as a sum of terms that involve x raised to a non-negative integer power. For example f(x) = x^2 is a polynomial, because it involves x raised to the power 2. Similarly, f(x) = 1 + x^2 + 2 x^3 is a polynomial, because all the terms being summed involve x raised to a non-negative power. Notice that the term 1, with no x in it, is x raised to the zero. In contrast f(x) = sqrt(2x) f(x) = x^-2 f(x) = e^x are all non-polynomials. B. Standard form A polynomial can be written in many ways, but one of the most useful is called standard form. Standard form just means that the terms are ordered from the highest power to the lowest power. Thus for example f(x) = 1 + x^2 + x^3 is not in standard form, while f(x) = x^3 + x^2 + 1 is the same polynomial in standard form. In general standard form looks like f(x) = a_n x^n + a_{n-1} x^{n-1} + a_{n-2} x^{n-2} + ... + a_2 x^2 + a_1 x + a_0 where the a's are numbers, called coefficients. The highest power of x present is called the degree of the polynomial, so for example f(x) = 2 x^4 + 2 x - 1 is a polynomial of degree 4. Polynomials of degree 1 are sometimes called linear, those of degree 2 are called quadratic, those of degree 3 are cubic, degree 4 is quartic, degree 5 is quintic, etc. The entire term containing the highest power present is called the leading term. In this example, the leading term is 2 x^4. Similarly, 2 is the leading coefficient. II. Solving and factoring polynomial equations Equations involving polynomials can sometimes be solved by factoring. To take an example x^4 - 5x^2 + 4 = 0 is an equation involving a polynomial. We can factor this to (x^2 - 4)(x^2 - 1) = 0 Each of these factors is a difference of squares and can be factored further: (x - 2)(x + 2)(x - 1)(x + 1) = 0 Clearly this can be solved by making any one of these four factors zero, so there are four solutions: x = -2, +2, -1, +1. The values of x for which a polynomial p(x) is equal to zero are called its roots. This illustrates a general principle: if you have an equation in the form p(x) = 0, where p(x) is a polynomial, if you can factor the polynomial than you can read off the answers. Note that, in our example, the polynomial was of degree 4, and we found 4 factors. This is another general principle: the number of factors is equal to the degree of the polynomial. It is easy to see why this should be, since every factor produces a term with one more power of x. The quadratic formula, we see, is a special case of this for polynomials of degree 2. It gives us the factors. It turns out that there are equivalent formulas for cubic polynomials and even quartic ones, but there is no such formula for polynomials of degree 5 or higher. It is possible to prove this rigorously. Thus we cannot in general factor high degree polynomials to find exact solutions. Instead, we are reduced to finding approximate solutions using computers. Note that factors are sometimes repeated. Thus for example the polynomial p(x) = x^2 - 4x + 4 factors to p(x) = (x-2)(x-2). This polynomial has two roots that are not distinct, they are both 2. This is called a root of multiplicity 2. We can turn this factoring around: given a set of roots, we can always construct the polynomial that has those roots, just by finding the corresponding factors. Thus if we seek a polynomial with roots 1, 2, and 3, we have p(x) = (x-1)(x-2)(x-3) = (x^2 - 3x + 2)(x - 3) = x^3 - 6x^2 + 11x - 6 III. Graphs A. Limiting behavior We can sketch out what graphs of polynomials look like by considering two features of their behavior: their roots and their leading terms. Let's take the example we just used: p(x) = x^3 - 6x^2 + 11x - 6 When x is very large, the x^3 term will be much larger than any of the other ones, because x^3 is always bigger than x^2 if we pick x sufficiently large. Thus at large values of x, the overall shape of the function will be determined by the leading term. The same goes for at x << 0, i.e. for very negative values of x. Thus the long-run behavior of polynomial functions only depends on their leading terms. In the example we have, x^3 is positive when x is positive, and negative when x is negative, so we expect the graph to be positive at large x and negative at small x. If the sign of the x^3 term were reversed it would be the other wa around: negative when x was large, positive when it was small. If we instead had p(x) ~ x^4, however, we notice that x^4 is always positive, whether x is positive or negative. Thus at large or small x the polynomial is always positive. If the sign were opposite, say p(x) ~ -2 x^4, then at large or small x the polynomial would always be negative. To sum up, for a polynomial p(x) whose leading term is a_n x^n, we have: - If n is even and a_n > 0, p(x) is positive and increasing toward infinity for both x >> 1 and x << 1 - If n is even and a_n < 0, p(x) is negative and decreasing toward negative infinity for both x >> 1 and x << 1 - If n is odd and a_n > 0, p(x) is positive and increasing toward infinity for both x >> 1, and negative and decreasing toward negative infinity for x << 1 - If n is odd and a_n < 0, p(x) is positive and increasing toward infinity for both x << 1, and negative and decreasing toward negative infinity for x >> 1 B. Zeros We also know what the graph looks like in the vicinity of roots of a polynomial. Suppose we have a root at x = k, so the function looks like p(x) = (x - k)^n . other factors, where n indicates the multiplicity of the root. Clearly p(x) = 0 at x = k, so the graph goes to zero at the root. The shape near the root depends on n. To see how it behaves, consider what happens if we pick x to be a little bigger than k or a little smaller than k. Suppose first that n is odd, for example n = 1. If x > k, then (x-k)^n is positive, and the sign of p(x) depends on whether the other factors are positive or negative. Conversely, for x < k, then (x-k)^n is negative, and the sign of p(x) is the opposite of whatever sign the other factors come out to have. This will be the same sign as at x > k, provided we choose x just barely to the left or right of k. In this case, the sign of p(x) is opposie for x slightly less than k than for x slightly greater than k, so p(x) passes through zero at x=k, and is positive on one side and negative on the other. Now suppose that n is even, so the root is of even multiplicity. In this case (x-k)^n is positive no matter whether we choose x just bigger or just smaller than k. As a result, we see that p(x) must hit the axis, but not pass through it. Instead, it "bounces off" the axis. Given this information, we can sketch a graph of a polynomial just given its zeros and leading coefficient. Example: sketch graph of (x - 3)(x + 1)^2(x-1) Practice problem: sketch graphs of (x - 4)^3 (x - 3)^4 (x - 2)^5