Algebra 2 Notes AII.7 Polynomials Part 2

Algebra 2 Notes AII.7 Polynomials Part 2 Mrs. Grieser Name: Date: Block: Zeros of a Polynomial Function So far: o If we are given a zero (or factor or solution) of a polynomial function, we can use division to simplify the polynomial and find other zeros What if we aren t given a zero (or factor or solution)? How do we guess what might be a zero (or factor or solution)? The Rational Zero Theorem If f(x) = anx n + an-1x n-1 + +a1x+a0 has integer coefficients, then every rational zero of f has the form: p q factor of constant term a factor of leading coefficient a 0 n CAUTION: Not every possibility named by the Rational Zero Theorem will be a zero of the polynomial function, but every rational zero of the polynomial will be in the list. NOTE 1: Include positive and negative factors in the list of possibilities. NOTE 2: It could be none of the possible zeros in the list are zeros of the function if the function has no rational zeros. Example 1: List all possible rational zeros of f(x) = 4x 4 x 3-3x 2 + 9x + 10. o List factors of 10 (ao): o List factors of 4 (an): o Create a list of possible zeros: Example 2: Find the zeros of f(x) = x 3-8x 2 + 11x + 20. o List factors of 20 (ao): o List factors of 1 (an): o Create a list of possible zeros: o Test them (how?) to see which might zeros o If a zero has been found, divide or factor and find others: You Try: a) Find possible rational zeros for: f(x) = x 3 + 9x 2 + 23x + 15 b) Find all real zeros for f(x) = x 3 4x 2 15x + 18

Algebra 2 Notes AII.7 Polynomials Part 2 Mrs. Grieser Page 2 Fundamental Theorem of Algebra Plain speak: A polynomial equation of degree n has n solutions (roots). Some of the solutions may be imaginary Repeated solutions are counted separately (multiplicity) Note that if imaginary or irrational numbers are zeros of a function, then their conjugates are also zeros. Example 1: How many solutions does the equation x 3 + 5x 2 + 4x + 20 = 0 have? The equation has 3 solutions. Explain What are they? o The rational zero theorem tells us possible rational solutions are: o Which possible rational zeros are solutions? o Divide and find other solutions: Fundamental Theorem of Algebra If f(x) is a polynomial of degree n where n > 0, then the equation f(x) = 0 has at least one solution in the set of complex numbers. Corollary: If f(x) is a polynomial of degree n where n > 0, then the equation f(x) = 0 has exactly n solutions provided each solution repeated twice is counted as two solutions, each solution repeated three times is counted as three solutions, and so on. o The solutions are Example 2: Find the zeros of f(x) = x 5 4x 4 + 4x 3 + 10x 2 13x 14 o The function has zeros o Possible rational zeros o Find the rational zeros and put function in factored form: o Find complex zeros: o List the zeros, including repeated zeros: You try: Find the zeros of f(x) = x 5 2x 4 + 8x 2-13x + 6

Algebra 2 Notes AII.7 Polynomials Part 2 Mrs. Grieser Page 3 Complex and Irrational Conjugates Complex Conjugates Theorem If f is a polynomial function with real coefficients, and a + bi is an imaginary zero of f, then a bi is also a zero of f. Irrational Conjugates Theorem Suppose f is a polynomial function with rational coefficients, and a and b are rational numbers such that b is irrational. If a + b is a zero of f, then a - b is also a zero of f. Example: Write a polynomial function f of least degree that has rational coefficients, a leading coefficient of 1, and 3 and 2 + 5 as zeros. What are the zeros? Write f(x) in factored form: Expand to create polynomial: Check by evaluating the polynomial at each zero: You try: Write a polynomial function f of least degree that has rational coefficients, a leading coefficient of 1, and zeros 3 and 3 i.

Algebra 2 Notes AII.7 Polynomials Part 2 Mrs. Grieser Page 4 Descartes Rule of Signs To further help find zeros, mathematician René Descartes found the following: Descartes Rule of Signs Let f(x) = anx n + an-1x n-1 + + a2x 2 + a1x + a0 be a polynomial function with real coefficients. The number of positive real zeros of f is either equal to the number of sign changes of f (x) or is less than that number by an even integer. If there is only one variation in sign, there is exactly one positive real zero. The number of negative real zeros of f is either equal to the number of sign changes of f (-x) or is less than that number by an even integer. If f (-x) has only one variation in sign, then f has exactly one negative real zero. Use the rule of signs to help eliminate possibilities found with the rational zero theorem. The maximum possible positive and negative zeros add to the degree of the polynomial. Example: Use Descartes' Rule of Signs to determine the number of real zeros of: f (x) = x 5 x 4 + 3x 3 + 9x 2 x + 5 1) Count sign changes in f(x): There are 4 sign changes 4 maximum possible positive real zeros Maximum 4 positive zeros, but there could also be 2 or 0 positive real zeros since some may be imaginary (count down by twos since theorem says even integer zeros) 2) Count sign changes in f(-x) Find f(-x) = (-x) 5 (-x) 4 + 3(-x) 3 + 9(-x) 2 (-x) + 5 = -x 5 x 4 3x 3 + 9x 2 + x + 5 Count sign changes in f(-x): There is 1 sign change exactly one negative real zero 3) Conclusions: There are 4, 2, or 0 real positive zeros, and 1 negative real zero You Try: Use Descartes rule of signs to determine number of real zeros or solutions: a) Find number of real solutions: 4x 7 + 3x 6 + x 5 + 2x 4 x 3 + 9x 2 + x + 1 = 0 b) Find number of real zeros: f(x) = x 5 + x 4 + 4x 3 + 3x 2 + x + 1 c) Find the possible number of real solutions: 2x 4 x 3 + 4x 2 5x + 3 = 0

Algebra 2 Notes AII.7 Polynomials Part 2 Mrs. Grieser Page 5 Graphing Polynomials, and Analyzing Graphs So far: use end behavior and table of values Now: put together what we have learned about finding zeros More information about what happens in the middle of the graph (between ends) Clarifications: what is the difference between zeros, x-intercepts, solutions, roots, and factors?? The following statements are equivalent: zeros k is a zero of a function (x-value on graph where a function s value is 0) x-intercepts solutions If k is an x-intercept, it is the x-value on the graph where the function crosses or touches the x-axis (very much like a zero) k is a solution to f(x) = 0 (value of x that makes the equation true) roots k is a root if it is a solution to f(x) = 0 factors (x k) is a factor of f(x) if f(x) (x k) has a remainder of 0 Generally speaking: functions have zeros, x-intercepts, and factors; equations have solutions and roots. Multiplicity of zeros is important. Zeros with EVEN multiplicity TOUCH the graph; zeros with ODD multiplicity CROSS the graph. Example 1: Graph 1 f ( x) ( x 3)( x 6 2 2) Fortunately the function is in factored form. If it wasn t, put it in factored form. Graph the zeros. Zeros of the function are Note multiplicity of zeros where does the graph touch the x-axis, and where does it cross it? Plot some points between and beyond the intercepts: x -2-1 0 1 2 y Determine end behavior: o What is the sign of the leading coefficient and the degree of the function? o As x -, f(x) o As x +, f(x) Draw graph by filling in curve through points plotted.

Algebra 2 Notes AII.7 Polynomials Part 2 Mrs. Grieser Page 6 Turning Points Location where function goes from increasing to decreasing (or vice versa). Turning points represent local maxima or local minima. In the previous example, approximately where is the local maximum? Where is the local minimum? A polynomial of degree n has at most n-1 turning points. If a polynomial has n distinct zeros, then it has exactly n-1 turning points. Determine intervals where functions increase or decrease by looking at turning points. o In the previous example, find the intervals where the function increases, and where it decreases. o Increases: o Decreases: In the example at right, in what intervals does the function increase? In the example at right, in what intervals does the function decrease? Where is it neither increasing nor decreasing? Note that finding the exact location of turning points becomes much easier in calculus! For now, estimate, or use the graphing calculator to help find exact locations of turning points. You try: a) Graph the function: f(x) = (x+1) 2 (x-1)(x-3) zeros multiplicity: which zeros cross and which zeros touch x- axis? plot some points in between and beyond zeros: End behavior: o As x -, f(x) o As x +, f(x) Draw graph

Algebra 2 Notes AII.7 Polynomials Part 2 Mrs. Grieser Page 7 b) Estimate the coordinates of each turning point and state whether the turning point is a local maximum or local minimum. Estimate all real zeros. Determine the least degree of the polynomial. Find the intervals the function increases and then find the intervals it decreases. Writing Polynomials from a Graph Write the cubic function shown in the graph at right. o Identify the zeros o Write the function in factored form: f(x) = o Find the value of a by substituting in a point that is not a zero: o Expand to polynomial form You try: Write a polynomial function whose graph passes through the given points: (-1, 0), (0, -12), (2, 0), (3, 0)