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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=k is a root, or solution, of the equation f(x)=0
  • k is a zero of function f
  • (k,0) is an x-intercept of the graph of y=f(x)
  • xk is a linear factor of f(x)
Let's understand this with the polynomial g(x)=(x3)(x+2), which can be written as g(x)=(x3)(x(2)).
First, we see that the linear factors of g(x) are (x3) and (x(2)).
If we set g(x)=0 and solve for x, we get x=3 or x=2. 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=3 and x=2 are solutions to g(x)=0, then 3 and 2 are zeros of the function g.
Finally, the x-intercepts of the graph of y=g(x) satisfy the equation 0=g(x), which was solved above. The x-intercepts of the equation are (3,0) and (2,0).

Check your understanding

1) What are the zeros of f(x)=(x+4)(x7)?
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2) The graph of function g crosses the x-axis at (2,0). What must be a root of the equation g(x)=0?
x=
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi

3) The zeros of function h are 1 and 3. Which of the following could be h(x)?
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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(x)=(x1)(x4)2, the number 4 is a zero of multiplicity 2.
Notice that when we expand f(x), the factor (x4) is written 2 times.
f(x)=(x1)(x4)(x4)
So in a sense, when you solve f(x)=0, you will get x=4 twice.
0=(x1)(x4)(x4)x1=0x4=0x4=0x=1x=4x=4
In general, if xk 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.

Check your understanding

4) Which zero of f(x)=(x3)(x1)3 has multiplicity 3?
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5) Which zero of g(x)=(x+1)3(2x+1)2 is a double zero?
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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(x)=(x1)(x4)2 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=1, it only touches the x-axis at x=4.
Let's look at the graph of a function that has the same zeros, but different multiplicities. For example, consider g(x)=(x1)2(x4). 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=1 and crosses the x-axis at x=4.
In general, if a function f has a zero of odd multiplicity, the graph of y=f(x) will cross the x-axis at that x value. If a function f has a zero of even multiplicity, the graph of y=f(x) will touch the x-axis at that point.

Check your understanding

6) In the graphed function, is the multiplicity of the zero 6 even or odd?
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7) Which is the graph of h(x)=x2(x3)?
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Challenge problem

8*) Which is the graph of f(x)=x3+4x24x?
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