Polynomial Functions: Learn It 3

Identifying Intercepts of Factored Polynomial Functions

When polynomial functions are written in factored form, finding their intercepts becomes straightforward. The factored form reveals the zeros directly, while the y-intercept can be found through substitution. This page will teach you to identify both types of intercepts efficiently.

factored form of a polynomial

A polynomial function in factored form is written as:

[latex]f(x) = a(x - r_1)(x - r_2)(x - r_3)\cdots(x - r_n)[/latex]

where:

  • [latex]a[/latex] is the leading coefficient
  • [latex]r_1, r_2, r_3, \ldots, r_n[/latex] are the zeros (roots) of the function

[latex]f(x) = 2(x - 3)(x + 1)(x - 5)[/latex]

This polynomial has zeros at [latex]x = 3[/latex], [latex]x = -1[/latex], and [latex]x = 5[/latex].

x-intercepts from Factored Form

How to: find x-intercepts from factored form:

  1. Set the function equal to zero: [latex]f(x) = 0[/latex]
  2. Use the Zero Product Property
  3. Solve each factor equal to zero
  4. The solutions are the x-intercepts

Find the x-intercepts of [latex]h(x) = -2(x - 5)(x + 3)(x - 2)[/latex].

Solution:

Step 1: Set the function equal to zero [latex]-2(x - 5)(x + 3)(x - 2) = 0[/latex]

Step 2: Apply the Zero Product Property Since [latex]-2 \neq 0[/latex], we focus on the factors containing [latex]x[/latex]:

  • [latex]x - 5 = 0[/latex] or
  • [latex]x + 3 = 0[/latex] or
  • [latex]x - 2 = 0[/latex]

Step 3: Solve each equation

  • [latex]x - 5 = 0 \Rightarrow x = 5[/latex]
  • [latex]x + 3 = 0 \Rightarrow x = -3[/latex]
  • [latex]x - 2 = 0 \Rightarrow x = 2[/latex]

Answer: The x-intercepts are [latex]x = 5[/latex], [latex]x = -3[/latex], and [latex]x = 2[/latex].

The leading coefficient affects the shape and direction of the graph but does not change the x-intercepts.

multiplicity

When a factor appears multiple times, we say it has multiplicity greater than 1.

Find the x-intercepts of [latex]f(x) = (x - 2)^2(x + 1)(x - 3)[/latex].

Solution:

Step 1: Identify the factors

  • [latex](x - 2)^2[/latex] – factor [latex](x - 2)[/latex] with multiplicity 2
  • [latex](x + 1)[/latex] – factor [latex](x + 1)[/latex] with multiplicity 1
  • [latex](x - 3)[/latex] – factor [latex](x - 3)[/latex] with multiplicity 1

Step 2: Find the zeros

  • [latex]x - 2 = 0 \Rightarrow x = 2[/latex] (multiplicity 2)
  • [latex]x + 1 = 0 \Rightarrow x = -1[/latex] (multiplicity 1)
  • [latex]x - 3 = 0 \Rightarrow x = 3[/latex] (multiplicity 1)

Answer: The x-intercepts are [latex]x = 2[/latex] (multiplicity 2), [latex]x = -1[/latex], and [latex]x = 3[/latex].

Finding y-intercepts

The y-intercept is the point where the graph crosses the y-axis, which occurs when [latex]x = 0[/latex].

How to: find the y-intercept:

  1. Substitute [latex]x = 0[/latex] into the function
  2. Evaluate the expression
  3. The result is the y-coordinate of the y-intercept

Find the y-intercept of [latex]f(x) = 2(x - 3)(x + 1)(x - 5)[/latex].