5-2 Polynomials, Linear Factors, & Zeros Today’s Objective: I can write and graph a polynomial function

Roots, Zeros & x-intercepts 𝑷 𝒙 = 𝒂 𝒏 𝒙 𝒏 + 𝒂 𝒏−𝟏 𝒙 𝒏−𝟏 +⋯+ 𝒂 𝟏 𝒙+ 𝒂 𝟎 Factor Theorem 𝑥−𝑏 is a linear factor of the polynomial 𝑃(𝑥) if and only if b is a zero of the polynomial function 𝑃(𝑥) Find the zeros Write the polynomial given the zeros: 2, −2, 3 𝑦= 𝑥−1 𝑥+2 (𝑥−3) 𝑦= 𝑥 3 +2 𝑥 2 −24𝑥 𝑦=(𝑥− )(𝑥− )(𝑥− ) 2 +2 3 𝑦=𝑥( 𝑥 2 +2𝑥−24) 1, −2, 3 𝑦=( 𝑥 2 −4)(𝑥−3) 𝑦=𝑥 𝑥+5 𝑥−7 𝑦=𝑥 𝑥−4 (𝑥+6) 𝑦= 𝑥 3 −3 𝑥 2 −4𝑥+12 0, −5, 7 0, 4, −6

Graphing with zeros 𝑓(𝑥)=𝑥(𝑥−4)(𝑥+3) Find and plot the zeros Sketch end behavior Pick easy midpoints between zeros to estimate turning point Zeros: 0, 4, −3 𝑓(−2)= −2(−2−4)(−2+3) =12 𝑓(2)= 2(2−4)(2+3) =−20

Zeros with Multiplicity 𝑓(𝑥)= (𝑥+2) 2 (𝑥−2)(𝑥−3) Find and plot the zeros Sketch end behavior Pick easy midpoints between zeros to estimate turning point 𝑓(𝑥)=(𝑥+2)(𝑥+2)(𝑥−2)(𝑥−3) Even multiplicity turns graph at zero Odd multiplicity pauses graph zero Zeros: −2, −2, 2, 3 𝑓(0)= (0+2) 2 (0−2)(0−3) =24 𝑓(2.5)= 2.5+2 2 (2.5−2)(2.5−3) =−5 W.S. 5-1/5-2 Polynomials, Linear Factors & Zeros