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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, 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.

Check your understanding

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?
Choose 1 answer:
Choose 1 answer:

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
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text

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

Check your understanding

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?
Choose 1 answer:
Choose 1 answer:

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?
Choose 1 answer:
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, 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.

Check your understanding

6) In the graphed function, is the multiplicity of the zero 6 even or odd?
Choose 1 answer:
Choose 1 answer:

7) Which is the graph of h, left parenthesis, x, right parenthesis, equals, x, squared, left parenthesis, x, minus, 3, right parenthesis?
Choose 1 answer:
Choose 1 answer:

Challenge problem

8*) Which is the graph of f, left parenthesis, x, right parenthesis, equals, minus, x, cubed, plus, 4, x, squared, minus, 4, x?
Choose 1 answer:
Choose 1 answer:

Want to join the conversation?

  • 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!!
    (15 votes)
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    • 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.
      (2 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?
    (5 votes)
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    • 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.
      (15 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?
    (6 votes)
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  • 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?
    (7 votes)
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  • aqualine ultimate style avatar for user shub112
    Using multiplity how can you find number of real zeros on a graph
    (3 votes)
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    • 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)
  • 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)
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  • duskpin ultimate style avatar for user Goat
    Why's it called a 'linear' factor?
    (2 votes)
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  • blobby green style avatar for user kslimba1972
    why the power of a polynomial can not be negative or in fraction?
    (3 votes)
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    • duskpin ultimate style avatar for user Elammen
      If you found the zeros for a factor of a polynomial function that contains a factor to a negative exponent, you’d find an asymptote for that factor with the negative power. For example: f(x)=(x+3)^2+(x-5)(x-3)^-1

      That (x-3)^-1 is equal to 1/(x-3), so it would be undefined at x=3 because that would make the denominator equal to 0, so a vertical asymptote is created at x=3.

      It doesn’t necessarily mean you can’t have negative powers in polynomial fractions, but it does mean that for negative exponents you will find an asymptote where the zero would be in the case of a positive exponent.
      (3 votes)
  • piceratops sapling style avatar for user devarakonda balraj
    how to find weather the graph is (odd or even)
    (2 votes)
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  • purple pi purple style avatar for user Advika
    I was doing SAT prep and this was the explanation for one of the questions-
    From the four factors of p(t), we see that the graph of y=p(t) must have at least two positive t-intercepts and at least two negative t‑intercepts.

    Only the first answer choice has at least two negative t‑intercepts. All of the other choices have one or less t‑intercepts.

    Can somebody explain this concept please?
    (2 votes)
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