Polynomials Functions 𝒇 𝒙 = 𝒂 𝒏 𝒙 𝒏 + 𝒂 𝒏−𝟏 𝒙 𝒏−𝟏 +…+ 𝒂 𝟎

Polynomial Expressions A polynomial is an algebraic expression that is the sum of terms involving variables that has whole number exponents In standard form, it is written in descending order of the degrees.

Standard Form of Polynomial Functions A polynomial function in standard form looks like 𝒇 𝒙 = 𝒂 𝒏 𝒙 𝒏 + 𝒂 𝒏−𝟏 𝒙 𝒏−𝟏 +…+ 𝒂 𝟎 where 𝑎 𝑖 are Reals coefficients and n are whole numbers. For example, 𝑓 𝑥 =2 𝑥 3 +4 𝑥 2 −5𝑥+7 represents a cubic function in standard form because the highest exponent is 3 followed by the quadratic term, linear term, and then the constant. In polynomials, the constant represents the y-intercept of the polynomial graph.

Examples of Polynomials Polynomials with 4 terms Monomial (1 term) Binomial ( 2 terms) Trinomial (3 terms) Polynomials with 4 terms 2 2𝑥+5 𝑥 2 +5𝑥 −4 𝑥 5 +5 𝑥 2 +3𝑥+1 − 2 𝑥 3 5 5 𝑥 3 −4 𝑥 2 9 𝑥 3 −4 𝑥 2 +5 𝑥 3 − 𝑥 2 +𝑥 −1

NOT Examples of Polynomials

Factored Form of Polynomials Polynomials may be written in factored form to explicitly show roots of the polynomials, A.K.A. the x-intercepts of the polynomial function. f(x)=a(x-x1)(x-x2)(x-x3) where x1, x2, and x3 are the x-intercepts of the function

A Turning Point of a polynomial is a point where there is a local max or a local min. polynomial of degree n can have at most n−1 turning points.

What is the least degree of the following polynomial functions? B) C)

End Behaviors Examine the sign and the degree of the leading term to know how polynomial graph behaves as it moves further left and further right.

Graphing Factored Form Directions: Find the x-intercepts by putting f(x)= 0 2) Find the y-intercept by putting x=0 3) Choose multiple x-values to find interpolating points 4) Plot all points and connect them with a smooth curve followed with the expected end-behaviors.

Graph f(x) = 2(x-3)(x+1)(x-1) Setting f(x) = 0, we get the x-intercepts 3,-1, and 1 Setting x=0, we get the y-intercept 6. X Y -1.5 -11.25 -1 -0.5 5.25 6 0.5 3.75 1 1.5 -3.75 2 -6 2.5 -5.25 3 3.5 11.25 4 30 Try: f(x) = x(x-3) (x+2)

Graphing Standard Form Directions: Find the y-intercept, which is the constant Factor the polynomial (if possible), then find the x-intercepts 3) Choose multiple x-values to find interpolating points 4) Plot all points and connect them with a smooth curve followed with the expected end-behaviors.

The y-intercept Graph 𝒚=−𝟐 𝒙 𝟒 +𝟓 Since the y-values shows symmetry about x=0 and there is only one term with the variable in the equation, the graph will behave similar to a parabola. X Y -2 -27 -1.5 -5.125 -1 3 -0.5 4.875 5 0.5 1 1.5 2

Graph f(x) = 2(x-3)(x+1)(x-1) Setting f(x) = 0, we get the x-intercepts 3,-1, and 1 Setting x=0, we get the y-intercept 6. X Y -1.5 -11.25 -1 -0.5 5.25 6 0.5 3.75 1 1.5 -3.75 2 -6 2.5 -5.25 3 3.5 11.25 4 30 Try: f(x) = x(x-3) (x+2)

Determining the Type of Polynomial Function by Table Values Make sure that X-Values are evenly spaced Are the differences constant? Add 1 to the degree of the polynomial. Continue to take another difference. NO Take the difference of the y-values YES Count the number of time each difference were made. That sum is the degree of the polynomial function.

Quadratic Function Example

Determine the Degree of the Polynomial X -4 -3 -2 -1 1 2 3 4 y -77 -38 -17 -8 -5 7 28 67 a) b) X 1 2 3 4 5 6 7 y 11 16 21 26 31