1. 5-2 Polynomials, Linear Factors, & Zeros
Today’s Objective:
I can write and graph a polynomial function

2. 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

3. 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

4. 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)(2.5−3)
=−5
W.S. 5-1/5-2 Polynomials, Linear Factors & Zeros