If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

Main content

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)?
Choose 1 answer:

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)?
Choose 1 answer:

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?
Choose 1 answer:

5) Which zero of g(x)=(x+1)3(2x+1)2 is a double zero?
Choose 1 answer:

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?
Choose 1 answer:

7) Which is the graph of h(x)=x2(x3)?
Choose 1 answer:

Challenge problem

8*) Which is the graph of f(x)=x3+4x24x?
Choose 1 answer:

Want to join the conversation?

  • duskpin ultimate style avatar for user Rutwik Pasani
    Why does the graph only touch the x axis at a zero of even multiplicity?
    (70 votes)
    Default Khan Academy avatar avatar for user
    • aqualine tree style avatar for user Judith Gibson
      I've been thinking about this for a while and here's what I've come up with.
      Let's say, for example, that f(x) = ( x - 4 ) ( x - 1 )^2.
      ( x - 4 ) is a root of odd multiplicity.
      Notice that when x < 4, ( x - 4 ) is negative,
      but when x > 4, ( x - 4 ) is positive.
      So, depending on the value of x, the sign of ( x - 4 ) changes, which in turn changes the sign of f(x).
      But also notice that for roots of even multiplicity [ ( x - 1 ) in this example], it doesn't matter what value of x is chosen. Once raised to their EVEN power, they will always be positive, so will not be able to change the sign of f(x).
      So, if f(x) is negative as it approaches a zero of EVEN multiplicity, then f(x) will remain negative after it passes that zero (and likewise if f(x) was positive, it would remain positive). In other words, it would just touch the x-axis and then have to "bounce" away in the same (positive or negative) direction.
      But if f(x) is negative (or positive) as it approaches a zero of ODD multiplicity, then f(x) will change sign --- in other words, the graph will cross through the x-axis instead of bouncing back.
      I hope this has been helpful and hasn't ended up confusing you!
      (279 votes)
  • female robot amelia style avatar for user Harsh Agrawal
    in the answer of the challenge question 8 how can there be 2 real roots . in total there are 3 roots as we see in the equation . but in the answer there are 2 real roots which will tell that there is only 1 imaginary root which does not exists. please help me . thanks in advance!!
    (18 votes)
    Default Khan Academy avatar avatar for user
    • stelly blue style avatar for user Kim Seidel
      There is no imaginary root. Sometimes, roots turn out to be the same (see discussion above on "Zeroes & Multiplicity"). That is what is happening in this equation. So, the equation degrades to having only 2 roots.
      If you factor the polynomial, you get factors of: -X (X - 2) (X - 2). You can see, 2 of the factors are identical.
      If you use these to solve for f(x) = 0, they create only 2 points: (0,0) and (2,0) because we have 2 identical factors that both create X=2.
      Hope this helps.
      (44 votes)
  • blobby green style avatar for user Kevin
    Why is Zeros of polynomials & their graphs important in the real world, when am i ever going to use this?
    (8 votes)
    Default Khan Academy avatar avatar for user
    • blobby green style avatar for user QUINN767
      It depends on the job that you want to have when you are older. School is meant to prepare students for any career path, including those that have to do with math. You might think now that you don't want a career with math, but you never know if you might decide to change your aspirations. When my mother was a child she hated math and thought it had no use, though later in life she actually went into a career that required her to have taken high math classes. You might use it later on! I'm grateful enough that I even have the opportunity to have such a nice education compared to developing countries where most citizens never make it to college.
      (23 votes)
  • hopper cool style avatar for user Timothy (Tikki) Cui
    For problem Check Your Understanding 6), if its "6", then why is it odd, not even?
    (10 votes)
    Default Khan Academy avatar avatar for user
    • aqualine tree style avatar for user Judith Gibson
      The question asks about the multiplicity of the root, not whether the root itself is odd or even.
      At a root of odd multiplicity, the graph will cross through the X-axis.
      At a root of even multiplicity, the graph will bounce off the X-axis and not go through it.
      (17 votes)
  • blobby green style avatar for user Michael Gomez
    In challenge problem 8, I don't know understand how we get the general shape of the graph, as in how do we know when it continues in the positive or negative direction. So for example, from left to right, how do we know that the graph is going to be generally decreasing?
    (9 votes)
    Default Khan Academy avatar avatar for user
  • aqualine ultimate style avatar for user emmat0802
    How would I solve f(x)=(2x-1)(x-5)?
    (4 votes)
    Default Khan Academy avatar avatar for user
    • stelly blue style avatar for user Kim Seidel
      You have a function with infinite solutions. It can be solved by using any input value for "x" and calculating "y".

      What where you asked to find? If you were asked to find the zeroes, they are the X-intercepts. Use y=0 and find x.
      (2x-1)(x-5) = 0
      The polynomial is already in factored form. So, use the zero product rule to split the factors apart
      2x-1=0 and x-5=0
      Solve these and you have the zeros.
      Hope this is what you meant by "sovle".
      (9 votes)
  • duskpin ultimate style avatar for user Bodhi
    Why's it called a 'linear' factor?
    (3 votes)
    Default Khan Academy avatar avatar for user
  • blobby green style avatar for user Anthony
    What if there is a problem like (x-1)^3 (x+2)^2 will the multiplicity be the addition of 3 and 2 or the highest exponent will be the multiplicity?
    (3 votes)
    Default Khan Academy avatar avatar for user
  • aqualine ultimate style avatar for user shub112
    Using multiplity how can you find number of real zeros on a graph
    (3 votes)
    Default Khan Academy avatar avatar for user
    • female robot grace style avatar for user loumast17
      So first you need the degree of the polynomial, or in other words the highest power a variable has. So if the leading term has an x^4 that means at most there can be 4 0s. There can be less as well, which is what multiplicity helps us determine. If a term has multiplicity more than one, it "takes away" for lack of a better term, one or more of the 0s.

      So for instance (x-1)(x+1)(x-2)(x+2) will have four zeros and each binomial term has a multiplicity of 1 Now, if you make one of them have a multiplicity of 2 that takes away one of the zeroes. so (x-1)(x-1)(x+2)(x-2), here there are two (x-1) terms so it has multiplicity 2, this means there is one less zero. So now there are only three zeroes at 1, 2 and -2. ALSO if a term has an even multiplicity it means it touches the x axis rather than crosses it.

      Let me know if that didn't help.
      (5 votes)
  • scuttlebug purple style avatar for user sophia.elizabeth.wright
    how can you figure out if the sign of f on the interval -1 < x < 4 is positive or negative?
    (4 votes)
    Default Khan Academy avatar avatar for user
    • cacteye blue style avatar for user Jerry Nilsson
      I guess you're talking about 𝑓(𝑥) = (𝑥 − 1)(𝑥 − 4)²,
      which has the zeros 𝑥 = 1 and 𝑥 = 4

      In expanded form we have 𝑓(𝑥) = 𝑥³ − 9𝑥² + 24𝑥 − 16

      Because the leading coefficient is positive and the degree of 𝑓 is odd,
      𝑓 will be negative for 𝑥 < 1

      Because the multiplicity of 𝑥 = 1 is odd,
      𝑓 will be positive over the interval 1 < 𝑥 < 4
      (3 votes)