Chapter 3: Polynomial and Rational Functions


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1 Chapter 3: Polynomial and Rational Functions 3.1 Polynomial Functions and Their Graphs A polynomial function of degree n is a function of the form P (x) = a n x n + a n 1 x n a 1 x + a 0 The numbers a 0, a 1,..., a n are called the coefficients of the polynomial. The number a 0 is called the constant coefficient (or constant term). The number a n, the coefficient of the highest power of x, is called the leading coefficient and the term a n x n is called the leading term. The end behavior of a polynomial is what happens to the graph as x becomes very large in either the positive or negative direction. In other words, what happens as x and as x. x is read as x approaches infinity x is read as x approaches negative infinity The end behavior of a general polynomial is determined by (1) the degree of the polynomial and (2) the sign of the leading coefficient. This gives us four cases: 1. Even degree, positive leading coefficient. (Think of x 2.) 2. Even degree, negative leading coefficient. (Think of x 2.) 3. Odd degree, positive leading coefficient. (Think of x 3.) 4. Odd degree, negative leading coefficient. (Think of x 3.) 1
2 Determine the end behavior of the following polynomials: P (x) = 2x x P (x) = x 5 + 5x 3 4x P (x) = (3x + 2) 2 (x 1) 3 (x + 4) We call a number c a zero of a polynomial P (x) if P (c) = 0. In other words, the zeros of a polynomial are the xintercepts. Example: Find the zeros of P (x) = x 2 + 9x + 20? The following 4 statements all mean the same thing for a polynomial P. 1. c is a zero of P. 2. x = c is a solution to the equation P (x) = x c is a factor of P. 4. x = c is an xintercept of the graph of P. Intermediate Value Theorem for Polynomials: If P is a polynomial function and P (a) and P (b) have opposite signs, then there exists at least one value c between a and b for which P (c) = 0, ie, there exists a value of c between a and b where c is a zero of P. What this tells us is that if I have two consecutive zeros, then the graph must lie entirely above or entirely below the xaxis between those zeros. 2
3 We call c a zero of multiplicity m when the factor x c occurs exactly m times in the factorization of P. What are the multiplicities of the zeros of P (x) = (x 1) 2 (x + 2) 3? If c is a zero of multiplicity m, there are 2 cases: 1. If m is odd, then the graph of P actually crosses the xaxis at this zero. 2. If m is even, then the graph of P does not completely cross the xaxis at this zero. It only touches it. To sketch the graph of a polynomial: 1. Factor the polynomial and determine the zeros (xintercepts) of the graph. 2. Plot the x and yintercepts. You may also want to plot some other points. 3. Determine the end behavior of the polynomial. 4. Sketch the graph that passes through the intercepts and any other points, making sure that the polynomial has the correct end behavior. Sketch the graph of P (x) = x 3 2x 2 + 8x. 3
4 Sketch the graph of P (x) = 2x 4 20x Sketch the graph of P (x) = (x 1) 2 (x + 2) 3. 4
5 3.2 Dividing Polynomials Long Division of Polynomials Dividing polynomials by long division is very similar to doing long division with numbers. There is still a dividend, divisor, quotient, and remainder. With polynomials, we always look first at the leading terms of the polynomials we re dividing. Examples Divide 6x 3 20x x 17 by 3x 4. We can write the answer in the form P (x) = D(x)Q(x) + R(x), where D(x) is the divisor, Q(x) is the quotient, and R(x) is the remainder. (R(x) must have degree less than D(x).) Another way to express this is: P (x) R(x) = Q(x) + D(x) D(x) Let P (x) = x 5 + x 4 2x 3 + x + 1 and D(x) = x 2 + x 1. Find Q(x) and R(x). 5
6 Synthetic division is a quick way to divide polynomials where the divisor is of the form x c. It allows us to just work with the coefficients instead of keeping track of the variables. Remember to include zeros for the powers of x that are not present. Examples. Use synthetic division to find the quotient and remainder 3x 3 12x 2 9x + 1 x 5 x 4 16 x + 2 Factor Theorem: c is a zero of P if and only if x c is a factor of P (x). Factor the polynomial P (x) = 3x 4 x 3 21x 2 11x + 6 completely if it is known that 1 3 of the polynomial. and 2 are zeros Find a polynomial of degree 4 that has zeros: 2, 0, 2, 4. Find a polynomial of degree 3 with integer coefficients that has zeros: 1 2, 1 3, and 1. 6
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