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## Algebra 2

### Unit 5: Lesson 2

Positive and negative intervals of polynomials

# Zeros of polynomials & their graphs

Learn about the relationship between the zeros, roots, and x-intercepts of polynomials. Learn about zeros multiplicities.

#### What you will learn in this lesson

When studying polynomials, you often hear the terms zeros, roots, factors and x-intercepts.
In this article, we will explore these characteristics of polynomials and the special relationship that they have with each other.

## Fundamental connections for polynomial functions

For a polynomial f and a real number k, the following statements are equivalent:
• x, equals, start color #01a995, k, end color #01a995 is a root, or solution, of the equation f, left parenthesis, x, right parenthesis, equals, 0
• start color #01a995, k, end color #01a995 is a zero of function f
• left parenthesis, start color #01a995, k, end color #01a995, comma, 0, right parenthesis is an x-intercept of the graph of y, equals, f, left parenthesis, x, right parenthesis
• x, minus, start color #01a995, k, end color #01a995 is a linear factor of f, left parenthesis, x, right parenthesis
Let's understand this with the polynomial g, left parenthesis, x, right parenthesis, equals, left parenthesis, x, minus, 3, right parenthesis, left parenthesis, x, plus, 2, right parenthesis, which can be written as g, left parenthesis, x, right parenthesis, equals, left parenthesis, x, minus, 3, right parenthesis, left parenthesis, x, minus, left parenthesis, minus, 2, right parenthesis, right parenthesis.
First, we see that the linear factors of g, left parenthesis, x, right parenthesis are left parenthesis, x, minus, start color #01a995, 3, end color #01a995, right parenthesis and left parenthesis, x, minus, left parenthesis, start color #01a995, minus, 2, end color #01a995, right parenthesis, right parenthesis.
If we set g, left parenthesis, x, right parenthesis, equals, 0 and solve for x, we get x, equals, start color #01a995, 3, end color #01a995 or x, equals, start color #01a995, minus, 2, end color #01a995. These are the solutions, or roots, of the equation.
A zero of a function is an x-value that makes the function value 0. Since we know x, equals, 3 and x, equals, minus, 2 are solutions to g, left parenthesis, x, right parenthesis, equals, 0, then start color #01a995, 3, end color #01a995 and start color #01a995, minus, 2, end color #01a995 are zeros of the function g.
Finally, the x-intercepts of the graph of y, equals, g, left parenthesis, x, right parenthesis satisfy the equation 0, equals, g, left parenthesis, x, right parenthesis, which was solved above. The x-intercepts of the equation are left parenthesis, start color #01a995, 3, end color #01a995, comma, 0, right parenthesis and left parenthesis, start color #01a995, minus, 2, end color #01a995, comma, 0, right parenthesis.

1) What are the zeros of f, left parenthesis, x, right parenthesis, equals, left parenthesis, x, plus, 4, right parenthesis, left parenthesis, x, minus, 7, right parenthesis?

2) The graph of function g crosses the x-axis at left parenthesis, 2, comma, 0, right parenthesis. What must be a root of the equation g, left parenthesis, x, right parenthesis, equals, 0?
x, equals

3) The zeros of function h are minus, 1 and 3. Which of the following could be h, left parenthesis, x, right parenthesis?

## Zeros and multiplicity

When a linear factor occurs multiple times in the factorization of a polynomial, that gives the related zero multiplicity.
For example, in the polynomial f, left parenthesis, x, right parenthesis, equals, left parenthesis, x, minus, 1, right parenthesis, left parenthesis, x, minus, 4, right parenthesis, start superscript, start color #aa87ff, 2, end color #aa87ff, end superscript, the number 4 is a zero of multiplicity start color #aa87ff, 2, end color #aa87ff.
Notice that when we expand f, left parenthesis, x, right parenthesis, the factor left parenthesis, x, minus, 4, right parenthesis is written start color #aa87ff, 2, end color #aa87ff times.
f, left parenthesis, x, right parenthesis, equals, left parenthesis, x, minus, 1, right parenthesis, start color #aa87ff, left parenthesis, x, minus, 4, right parenthesis, left parenthesis, x, minus, 4, right parenthesis, end color #aa87ff
So in a sense, when you solve f, left parenthesis, x, right parenthesis, equals, 0, you will get x, equals, 4 twice.
\begin{aligned}0&=(x-1)\purpleC{(x-4)(x-4)}\\ \\ &x-1=0\qquad x-4=0\qquad x-4=0\\\\ &x=1\qquad \qquad \purpleC{x=4}\qquad \qquad \purpleC{x=4} \end{aligned}
In general, if x, minus, k occurs m times in the factorization of a polynomial, then k is a zero of multiplicity m. A zero of multiplicity 2 is called a double zero.

4) Which zero of f, left parenthesis, x, right parenthesis, equals, left parenthesis, x, minus, 3, right parenthesis, left parenthesis, x, minus, 1, right parenthesis, cubed has multiplicity 3?

5) Which zero of g, left parenthesis, x, right parenthesis, equals, left parenthesis, x, plus, 1, right parenthesis, cubed, left parenthesis, 2, x, plus, 1, right parenthesis, squared is a double zero?

## The graphical connection

The multiplicity of a zero is important because it tells us how the graph of the polynomial will behave around the zero.
For example, notice that the graph of f, left parenthesis, x, right parenthesis, equals, left parenthesis, x, minus, 1, right parenthesis, left parenthesis, x, minus, 4, right parenthesis, squared behaves differently around the zero 1 than around the zero 4, which is a double zero.
Specifically, while the graphs crosses the x-axis at x, equals, 1, it only touches the x-axis at x, equals, 4.
Let's look at the graph of a function that has the same zeros, but different multiplicities. For example, consider g, left parenthesis, x, right parenthesis, equals, left parenthesis, x, minus, 1, right parenthesis, squared, left parenthesis, x, minus, 4, right parenthesis. Notice that for this function 1 is now a double zero, while 4 is a single zero.
Now we see that the graph of g touches the x-axis at x, equals, 1 and crosses the x-axis at x, equals, 4.
In general, if a function f has a zero of odd multiplicity, the graph of y, equals, f, left parenthesis, x, right parenthesis will cross the x-axis at that x value. If a function f has a zero of even multiplicity, the graph of y, equals, f, left parenthesis, x, right parenthesis will touch the x-axis at that point.

6) In the graphed function, is the multiplicity of the zero 6 even or odd?