Lesson 3 – Polynomial Functions in Factored Form
Recall:
Graph the function 𝑓(𝑥) = 2(𝑥 − 3)(𝑥 + 1) using the zeros and y-intercept.
Factored Form
The factored form of a Polynomial Function is:
𝑓(𝑥) = 𝑎(𝑥 − 𝑥1 )(𝑥 − 𝑥2 )(𝑥 − 𝑥3 ) ⋯ (𝑥 − 𝑥𝑛 )
where, 𝑥1 , 𝑥2 , 𝑥3 , … , 𝑥𝑛 are the zeros of f(x) and a is the leading coefficient.
Single Root
𝑓(𝑥) = 𝑎(𝑥 − 𝑥1 )(𝑥 − 𝑥2 )(𝑥 − 𝑥3 )
Example 1:
Double Root
𝑓(𝑥) = 𝑎(𝑥 − 𝑥1
)2 (𝑥
Triple Root
− 𝑥2 )
𝑓(𝑥) = 𝑎(𝑥 − 𝑥1 )𝟑 (𝑥 − 𝑥2 )
State the degree, leading coefficient and end behavior of the function:
𝑓(𝑥) = −2(𝑥 − 2)3 (𝑥 + 3)(𝑥 − 1)
Example 2:
Sketch the graph of 𝑓(𝑥) = −(𝑥 + 1)(𝑥 − 1)(𝑥 − 2)2 using the zeros, the y
intercepts and surrounding points.
Example 3:
Determine the equation of the polynomial function in factored form.
Summary
The zeros of a polynomial function 𝑦 = 𝑓(𝑥) are the same as the roots of the related
polynomial equation 𝑓(𝑥) = 0
To determine the equation of the polynomial function in factored form, follow these
steps
 Substitute the zeros 𝑥1 , 𝑥2 , 𝑥3 , … , 𝑥𝑛 into the general equations of the appropriate
family (same degree) of polynomial functions of the form
𝑓(𝑥) = 𝑎(𝑥 − 𝑥1 )(𝑥 − 𝑥2 )(𝑥 − 𝑥3 ) ⋯ (𝑥 − 𝑥𝑛 )
 Substitute the coordinates of an additional point for x and y, and solve for a to
determine the equation