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 xx-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 ff and a real number kk, the following statements are equivalent:
  • x=kx=\tealD k is a root, or solution, of the equation f(x)=0f(x)=0
  • k\tealD k is a zero of function ff
  • (k,0)(\tealD k,0) is an xx-intercept of the graph of y=f(x)y=f(x)
  • xkx-\tealD k is a linear factor of f(x)f(x)
Let's understand this with the polynomial g(x)=(x3)(x+2)g(x)=(x-3)(x+2), which can be written as g(x)=(x3)(x(2))g(x)=(x-3)(x-(-2)).
First, we see that the linear factors of g(x)g(x) are (x3)(x-\tealD3) and (x(2))(x-(\tealD{-2})).
If we set g(x)=0g(x)=0 and solve for xx, we get x=3x=\tealD3 or x=2x=\tealD{-2}. These are the solutions, or roots, of the equation.
A zero of a function is an xx-value that makes the function value 00. Since we know x=3x=3 and x=2x={-2} are solutions to g(x)=0g(x)=0, then 3\tealD3 and 2\tealD{-2} are zeros of the function gg.
Finally, the xx-intercepts of the graph of y=g(x)y=g(x) satisfy the equation 0=g(x)0=g(x), which was solved above. The xx-intercepts of the equation are (3,0)(\tealD3,0) and (2,0)(\tealD{-2},0).

Check your understanding

1) What are the zeros of f(x)=(x+4)(x7)f(x)=(x+4)(x-7)?
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We know that if xkx-\tealD k is a linear factor of f(x)f(x), then k\tealD k is a zero of function ff.
We can rewrite f(x)=(x+4)(x7)f(x)=(x+4)(x-7) as f(x)=(x(4))(x7)f(x)=(x-(\tealD{-4}))(x-\tealD7). Therefore, the zeros of ff are 4\tealD{-4} and 7\tealD 7.
2) The graph of function gg crosses the xx-axis at (2,0)(2,0). What must be a root of the equation g(x)=0g(x)=0?
x=x=
  • Your answer should be
  • an integer, like 66
  • a simplified proper fraction, like 3/53/5
  • a simplified improper fraction, like 7/47/4
  • a mixed number, like 1 3/41\ 3/4
  • an exact decimal, like 0.750.75
  • a multiple of pi, like 12 pi12\ \text{pi} or 2/3 pi2/3\ \text{pi}

If (k,0)(\tealD k,0) is an xx-intercept of the graph of a function ff, then we know that x=kx=\tealD k is a root, or solution, of the equation f(x)=0f(x)=0.
Since the graph of gg crosses the xx-axis at (2,0)(\tealD2,0), then x=2x=\tealD2 is a root of g(x)=0g(x)=0.
3) The zeros of function hh are 1-1 and 33. Which of the following could be h(x)h(x)?
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We know that if k\tealD k is a zero of function ff, then xkx-\tealD k is a linear factor of f(x)f(x).
Since the zeros of function hh are 1-1 and 33, we know that (x(1))(x-(-1)) and (x3)(x-3) must be factors of hh.
Simplifying, we see that the factors of h(x)h(x) must be (x+1)(x+1) and (x3)(x-3). The only option with these factors is: h(x)=(x+1)(x3)h(x)=(x+1)(x-3)

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)2f(x)=(x-1)(x-4)^\purpleC{2}, the number 44 is a zero of multiplicity 2\purpleC{2}.
Notice that when we expand f(x)f(x), the factor (x4)(x-4) is written 2\purpleC{2} times.
f(x)=(x1)(x4)(x4)f(x)=(x-1)\purpleC{(x-4)(x-4)}
So in a sense, when you solve f(x)=0f(x)=0, you will get x=4x=4 twice.
0=(x1)(x4)(x4)x1=0x4=0x4=0x=1x=4x=4\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 xkx-k occurs mm times in the factorization of a polynomial, then kk is a zero of multiplicity mm. A zero of multiplicity 22 is called a double zero.

Check your understanding

4) Which zero of f(x)=(x3)(x1)3f(x)=(x-3)(x-1)^3 has multiplicity 33?
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f(x)=(x3)(x1)3f(x)=(x-3)(x-1)^\tealD{3}
Notice that (x1)(x-1) appears three times in the factorization of f(x)f(x). The zero that relates to this factor is 11. Therefore 11 has a multiplicity of 33.
5) Which zero of g(x)=(x+1)3(2x+1)2g(x)=(x+1)^3(2x+1)^2 is a double zero?
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g(x)=(x+1)3(2x+1)2g(x)=(x+1)^3(2x+1)^\tealD2
Notice that (2x+1)(2x+1) appears twice in the factorization of g(x)g(x). The zero that relates to this factor is 12-\dfrac12.
2x+1=02x=1x=12\begin{aligned}2x+1&=0\\ \\ 2x&=-1\\ \\ x&=-\dfrac12 \end{aligned}
Therefore 12-\dfrac12 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(x)=(x1)(x4)2f(x)=(x-1)(x-4)^2 behaves differently around the zero 11 than around the zero 44, which is a double zero.
Specifically, while the graphs crosses the xx-axis at x=1x=1, it only touches the xx-axis at x=4x=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)g(x)=(x-1)^2(x-4). Notice that for this function 11 is now a double zero, while 44 is a single zero.
Now we see that the graph of gg touches the xx-axis at x=1x=1 and crosses the xx-axis at x=4x=4.
In general, if a function ff has a zero of odd multiplicity, the graph of y=f(x)y=f(x) will cross the xx-axis at that xx value. If a function ff has a zero of even multiplicity, the graph of y=f(x)y=f(x) will touch the xx-axis at that point.

Check your understanding

6) In the graphed function, is the multiplicity of the zero 66 even or odd?
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The graph of the polynomial crosses the xx-axis at (6,0)(6,0). Therefore, 66 is a zero with odd multiplicity.
7) Which is the graph of h(x)=x2(x3)h(x)=x^2(x-3)?
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The function has two zeros, 00 and 33, and so the graph of y=h(x)y=h(x) must have xx-intercepts at (0,0)(0,0) and (3,0)(3,0).
Because xx occurs twice in the factorization of hh, the related zero, 00, has a multiplicity of 22. This is even, and so the graph of hh will touch the xx-axis at x=0x=0.
Because x3x-3 occurs once in the factorization of hh, the related zero, 33, has a multiplicity of 11. This is odd, and so the graph of hh will cross the xx-axis at x=3x=3.
The only graph that reflects this is BB.

Challenge problem

8*) Which is the graph of f(x)=x3+4x24xf(x)=-x^3+4x^2-4x?
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Notice that this polynomial is in standard form, and so we must factor the polynomial in order to find the zeros.
f(x)=x3+4x24x=x(x24x+4)=x(x2)(x2)=x(x2)2\begin{aligned} f(x)&=-x^3+4x^2-4x \\ \\&= -x(x^2-4x+4) \\ \\ &= -x(x-2)(x-2)\\ \\&=-x(x-2)^2 \end{aligned}
From this we see that 00 is a zero with odd multiplicity and 22 is a zero with even multiplicity. So the graph of ff will cross the xx-axis at (0,0)(0,0) and touch the xx-axis at (2,0)(2,0).
The only option to reflect this is graph DD
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