Polynomial Functions Some Terminology: Monomial: Real number, or variable, or product of a real number and variables with exponents which are whole numbers. The degree of the monomial (in one variable) is the exponent on the variable. Polynomial: A monomial or sum of monomials. The degree of the polynomial is the highest degree among its monomial terms. Polynomial Function: Say, of “𝑥”, would be the a polynomial with terms involving the variable “𝑥”. For example: 𝑓 𝑥 =3 𝑥 3 +2𝑥+4 would be a polynomial function of 𝑥 of degree 3, also called a cubic function of 𝑥.

Standard Form for a Polynomial Function Terms are arranged in descending order of degree: 𝑃 𝑥 = 𝑎 𝑛 𝑥 𝑛 + 𝑎 𝑛−1 𝑥 𝑛−1 +…+ 𝑎 1 𝑥+ 𝑎 0 Names of polynomials by number of terms: Monomial: polynomial with one term Binomial: polynomial with two terms Trinomial: polynomial with three terms Names of polynomial functions by highest degree: Name Constant Linear Quadratic Cubic Quartic Quintic Degree 1 2 3 4 5 Leading Term 𝑎 0 𝑎 1 𝑥 𝑎 2 𝑥 2 𝑎 3 𝑥 3 𝑎 4 𝑥 4 𝑎 5 𝑥 5

Behavior of Polynomial Functions Important characteristics of functions typically include: End behavior: How function 𝑓 𝑥 behaves as 𝑥→±∞ ( → means approaches) Turning Points: Places where the function changes direction (that is, where the slope changes sign) An n-degree polynomial has, at most, (n-1) turning points. A polynomial of odd degree has an even number of turning points. A polynomial of even degree has an odd number of turning points.

Some Exercises Write each polynomial in standard form, classify it by degree and by number of terms: 3x + 9x2 + 5 4x – 6x2 + x4 + 10x2 – 12 For each function, find the end behavior, then using a graphing calculator, find the turning points, and the intervals on which the function is increasing and decreasing. 𝑦=4 𝑥 4 +6 𝑥 3 −𝑥 𝑦=− 𝑥 2 +2𝑥 𝑦= 𝑥 3 𝑦=− 𝑥 3 +2𝑥

How Can We Find the Degree of a Polynomial from Data? Given a set of outputs 𝑦 for consecutive and evenly spaced (differing by a constant) inputs 𝑥 for a polynomial, find its degree. For degree 1, the first differences of the data will be the same. For degree 2, the second differences of the data will be the same…. For degree n, the nth differences of the data will be the same. Find the order of the following polynomials given the data. x -3 -2 -1 1 2 3 y -7 5 11 9 x -3 -2 -1 1 2 3 y 23 -16 -15 -10 -13 -12 29

Finding Degree of Polynomial from Data Find the order of the following polynomials given the data. x -3 -2 -1 1 2 3 y -7 5 11 9 x -3 -2 -1 1 2 3 y 23 -16 -15 -10 -13 -12 29

Linear Factors and Zeros of a Polynomial We can solve some polynomial equations: 𝑎 𝑛 𝑥 𝑛 + 𝑎 𝑛−1 𝑥 𝑛−1 +…+ 𝑎 1 𝑥+ 𝑎 0 = 0 by finding linear factors: 𝑥− 𝑟 1 𝑥− 𝑟 2 ⋯ 𝑥− 𝑟 𝑛 = 0 Some terminology and properties: For a polynomial function: 𝑃 𝑥 = 𝑎 𝑛 𝑥 𝑛 + 𝑎 𝑛−1 𝑥 𝑛−1 +…+ 𝑎 1 𝑥+ 𝑎 0 If 𝑥−𝑏 is a linear factor of 𝑃 𝑥 , then b is a zero of P(x). b is a root of the equation: P(x) = 0. b is an x-intercept of the graph of y = P(x). Theorem: The expression 𝑥−𝑏 is a factor of a polynomial if and only if the value b is a zero of the related polynomial function.

Some Exercises For y = (x+2)(x-1)(x-3), Find the zeros, graph the function, and put it in standard form. First find the zeros Find y for values of x between zeros Identify the end behavior Sketch the function Multiply out the factors, combine like terms, and rearrange terms to put in standard form.

More Exercises A cubic polynomial function has zeros: -2, 2, 3. Write it in standard form. A quartic polynomial function has zeros -2, -2, 2, 3. Write it in standard form.

Multiplicity of Zeros The quartic polynomial function of the last example can be written: f(x) = (x+2)2 (x-2)(x-3) We say it has a “multiple zero” and that -2 is a zero with a multiplicity of 2. A function with a zero of multiplicity n behaves like a function of degree n near that zero. So: f(x) behaves like: a quadratic near x =-2 (zero has multiplicity of 2) a line near x =2 and x = 3 (zeros each with multiplicity of 1)

Exercise For f(x) = x4 - 2x3 - 8x2 Find the zeros and their multiplicities. How does the graph behave near these zeros? Repeat this exercise for: f(x) = x3 - 4x2 + 4x

Relative Maxima and Minima At a Turning Point, the function achieves an extreme value relative to neighboring points. We call this a Relative (or local ) Maximum or Minimum. More generally these are “relative extrema”. If a function, at a point, achieves its lowest or highest value over its entire range, we call this a GLOBAL maximum or minimum. A whole field of optimization is devoted to trying to find the global extrema of functions without getting “trapped” or deceived by local extrema.

Modeling with a Polynomial A digital box camera maximizes the volume while keeping the sum of dimensions at 6 inches. Also, the length must be 1.5 times the height. So what should each dimension be? Write the Volume as a function of length, width, and height. Write equations for the two constraints in the problem. Using the constraint equations, rewrite the volume as a function of one variable (say, the height). Plot the function on the TI-nspire. Then choose : Menu->Analyze Graph(6)->Maximum(3). Use the touch pad to choose lower and upper bounds bracketing the relative maximum. Step 4 will give the height and maximum volume. Calculate the length and width using the constraint equations.

Try this problem Page 294: Problem 39.

How can we divide Polynomials? Always start with all polynomials in standard form. Include “zero” terms (terms where zero multiplies a lower power of x) for clarity. Divide: 6x2 + 7x +2 by 2x+1 4x2 + 23x – 16 by x+5 (Note: This example will leave a remainder) 8x3 – 1 by (x - ½ ) How can this example be used to find the zeros of 8x3 – 1 ?

Synthetic Division Streamlines the division process. Valid for dividing polynomials by linear factors (x-a) only. Write the coefficients of all the terms in descending order from the left, including zero terms. In a bracket to the left, write the “a” from the (x-a) factor. Bring down the first (leftmost) coefficient. Multiply the coefficient by the divisor (a) and add it to the next coefficient bringing down the result. Continue the process, working your way to the right, until you reach the last coefficient. The result brought down from the last coefficient will be the remainder.

Synthetic Division Exercises Try these: x3 – 14x2 + 51x -54 by x-2 x3 + 7x2 - 38x – 240 by x+5

Hwk 22 (due Tues 1/20) Page 285-286: 12, 24, 34, 40 Page 293-295: 25, 30, 44 Page 308-310: 12, 26

Remainder Theorem If you divide a Polynomial P(x) of degree n ≥ 1 by (x-a), then the remainder is P(a). Proving it: Let P(x) = the polynomial Let D(x) = (x-a) = the divisor Let Q(x) = the quotient. Let R = Remainder = a real number since it must be of lower degree than (x-a) which is degree 1. So: P(x) = D(x) Q(x) + R or: P(x) = (x-a) Q(x) + R Substituting “a” for “x” in the above equation: P(a) = (a-a) Q(a) + R = R

Using the Remainder Theorem Find the remainder when f(x) = 3x2 + 7x – 20 is divided by x-2 x+1 x+4 Check, using synthetic division.

Solving Polynomial Equations Methods we’ll explore: Graphing: Versatile. Factoring: Relies on recognizing familiar patterns in polynomial expressions; won’t always work. Dividing Polynomials: If you can find one root; say x = r1, then dividing the polynomial by the linear factor (x – r1) will reduce its degree by 1. Once you get to a quadratic, you can use the quadratic formula.

A Bag of Tricks See Page 297 of your text.

Solving Polynomial Equations x4 – 3x2 = 4 (Hint: try a “change of variables”) x3 = 1 x4 = 16 x3 = 8x – 2x2 x(x2 + 8) = 8(x+1)

An Interesting Graphing Method Solve: x3 + 5 = 4x2 + x The equation is satisfied when the Right Hand Side (RHS) equals the Left Hand Side (LHS). So: Plot f1(x) = x3 + 5 and f2(x) = 4x2 + x simultaneously and see where they intersect. You’ll have to zoom out to get all three roots. Hit menu-> analyze graph-> intersect. Or: conventionally: plot: f(x) = x3 – 4x2 –x + 5 then Hit menu-> analyze graph-> zero Use trace function to locate minima and maxima too.

Problems that come up Everyday 1. Stacy, Una, and Amir were all born on July 4th . Stacy is one year younger than Una. Una is two years younger than Amir. On July 4th, 2010 the product of their ages was 2300 more than the sum of their ages. How old was each person on that day? (Note: How come we have to leave units out here?) 2. Find three consecutive integers whose product is 480 more than their sum.

Hwk 23 Math Hwk 23 Algebra2Trig Page 300-302: 16, 20, 37 (set up an equation in one variable for solving on a graphing calculator. Solve it with a graphing calculator or at before class on Wednesday.), 64. Page 308: 49-52. Due Wednesday 1/21

Theorems about Roots of Polynomials 1. Rational Root Theorem: Let 𝑃 𝑥 = 𝑎 𝑛 𝑥 𝑛 + 𝑎 𝑛−1 𝑥 𝑛−1 +…+ 𝑎 1 𝑥+ 𝑎 0 be a polynomial with integer coefficients. Then there are a limited number of possible roots to the equation: 𝑃 𝑥 =0. a) Integer roots must be factors of 𝑎 0 . b) Rational roots must have a reduced form: 𝑝 𝑞 where p is an integer factor of 𝑎 0 and q is an integer factor of 𝑎 𝑛 . Ex: 21x2 + 29x + 10 = 0 Simply form all possible candidates for roots and then test them using synthetic division until you find a root.

Using the Rational Root Theorem Strategy: Use synthetic division with candidate roots until you find a root (no remainder). A success leaves you with a polynomial with lower degree. Repeat until you get a quadratic. Then use the quadratic formula. Examples: Find the rational roots of: 3x3 + 4x2 - 5x - 2 = 0 x2 - 2 = 0 (hmm....the theorem does NOT guarantee an answer.

Rational Root Theorem: Another Example What are the roots of: 2x3 + x2 - 7x – 6 = 0?

The Conjugate Root Theorem There are Two Parts to this theorem: Irrational Root Theorem: If P(x) is a polynomial with rational coefficients, then irrational roots of P(x) = 0 occur in conjugate pairs: 𝑎+ 𝑏 and 𝑎− 𝑏 , where 𝒂 and 𝒃 are rational. Imaginary Root Theorem: If P(x) is a polynomial with real coefficients, then complex roots of P(x) = 0 occur in complex conjugate pairs: 𝑎+𝑏𝑖 and 𝑎−𝑏𝑖 where 𝒂 and 𝒃 are real. Where does it come from? If you get enough factors to reduce a polynomial down to a quadratic, the last two solutions will come in a conjugate pair from: 𝑥= −𝑏± 𝑏 2 −4𝑎𝑐 2𝑎 .

What are the Conjugates of the Following Terms? 1+3𝑖 4 − 7 −2−9𝑖 15+ 10

Using the Conjugate Root Theorem A quartic Polynomial function P(x) has rational coefficients with roots 2 and 1+𝑖 . a) What are the other roots? b) What is the polynomial function? c) Given the polynomial we found in (b), can we use the rational root theorem to go the other way and get a root? 2) A cubic polynomial P(x) has real coefficients with roots: 3−2𝑖 and 5 2 .

More Using the Conjugate Root Theorem A third degree polynomial function P(x) has roots for P(x) = 0 of -4 and 2i. Find P(x). A quartic polynomial function P(x) has roots for P(x) = 0 of 2- 3i , 8, and 2. Find P(X).

Descartes’ Rule of Signs Let P(x) be a polynomial in standard form with real coefficients. The number of positive real roots equals the number of sign changes in P(x), or is less than that by an even number. The number of negative real roots equals the number of sign changes in P(-x), or is less than that by an even number.

Using Descartes’ Rule of Signs What does the rule tell you about: x3 – x2 + 1 = 0 2x4 – x3 + 3x2 - 1 = 0 2x5 -3x2 – 3x + 6

Hwk 24 Due Wednesday Math Hwk 24 Due Wed Page 316-317: 14, 18, 25, 30, 32, 38, 40

Fundamental Theorem of Algebra If P(x) is a polynomial function of degree n ≥ 1, then P(x) = 0 has exactly n roots, including multiple and complex roots. Steps to find zeros of a polynomial: Use a graphing calculator to find real roots Factor out linear factors of the form (x-a) using synthetic division. Use the Quadratic Formula to find complex roots. Example: Find all zeros of: 𝑓 𝑥 = 𝑥 4 + 𝑥 3 −7 𝑥 2 −9𝑥−18

The Binomial Theorem If you raise a binomial to the nth power, you’ll get an expression of this form: (a + b)n = P0 an + P1 an-1b + P2 an-2b2 + … + Pn-2 a2bn-2 +Pn-1 abn-1 + Pn bn Where the “P”s are constant coefficients given by the nth row of “Pascal’s Triangle”: Examples: Pascal’s Triangle Row (a+b)0 = 1 1 0 (a+b)1 = 1a+1b 1 1 1 (a+b)2 = 1a + 2ab + 1b 1 2 1 2 (a+b)3 = 1a3+3a2b+3ab2+1b3 1 3 3 1 3 (a+b)4 = 1a4+4a3b+6a2b2+4ab3+1b4 1 4 6 4 1 4

Applying the Binomial Theorem Expand the following using the Binomial Theorem: (x + a)3 (x - 2)5 (2x + 4)2 (3a - 2)3 The side length of a cube is (x2 - 1 2 ). Express the volume of the cube in standard form.

Transformations Revisited We’ve seen how to transform absolute value functions: Parent Function: 𝑓 𝑥 =|𝑥| Into the Transformed Function: 𝑦=𝑎 𝑥−ℎ +𝑘 And how to transform quadratic functions: Parent Function: 𝑓 𝑥 = 𝑥 2 Into the Transformed Function: 𝑦=𝑎 𝑥−ℎ 2 +𝑘 In the same way we can transform general parent functions: 𝑓(𝑥) into transformed functions: 𝑦=𝑎𝑓 𝑥−ℎ +𝑘

Examples For the parent function: y=x3, what function represents : A compression by ½, a reflection across the x-axis, a shift to the right by 3, and a vertical shift upwards by 2? A vertical stretch by 2, a shift to the left by 3, and a vertical shift down by 4?

Hwk 25 Page 322: 18, 39 (state each value of x at which a bridge is needed.) Pages: 328-330: 10, 36, 44, 48, 52 (You must use the binomial theorem for all of these problems; don’t multiply the binomials long hand.) Page: 343: 12 Due Friday