Zeros and multiplicity | Polynomial functions (article) | Khan Academy (2024)

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

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  • Rutwik Pasani

    8 years agoPosted 8 years ago. Direct link to Rutwik Pasani's post “Why does the graph only t...”

    Why does the graph only touch the x axis at a zero of even multiplicity?

    (70 votes)

    • Judith Gibson

      8 years agoPosted 8 years ago. Direct link to Judith Gibson's post “I've been thinking about ...”

      Zeros and multiplicity | Polynomial functions (article) | Khan Academy (4)

      Zeros and multiplicity | Polynomial functions (article) | Khan Academy (5)

      Zeros and multiplicity | Polynomial functions (article) | Khan Academy (6)

      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!

      (276 votes)

  • Harsh Agrawal

    8 years agoPosted 8 years ago. Direct link to Harsh Agrawal's post “in the answer of the chal...”

    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)

    • Kim Seidel

      8 years agoPosted 8 years ago. Direct link to Kim Seidel's post “There is no imaginary roo...”

      Zeros and multiplicity | Polynomial functions (article) | Khan Academy (10)

      Zeros and multiplicity | Polynomial functions (article) | Khan Academy (11)

      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.

  • Kevin

    7 years agoPosted 7 years ago. Direct link to Kevin's post “Why is Zeros of polynomia...”

    Why is Zeros of polynomials & their graphs important in the real world, when am i ever going to use this?

    (8 votes)

    • QUINN767

      6 years agoPosted 6 years ago. Direct link to QUINN767's post “It depends on the job tha...”

      Zeros and multiplicity | Polynomial functions (article) | Khan Academy (15)

      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)

  • Timothy (Tikki) Cui

    7 years agoPosted 7 years ago. Direct link to Timothy (Tikki) Cui's post “For problem Check Your Un...”

    For problem Check Your Understanding 6), if its "6", then why is it odd, not even?

    (10 votes)

    • Judith Gibson

      7 years agoPosted 7 years ago. Direct link to Judith Gibson's post “The question asks about t...”

      Zeros and multiplicity | Polynomial functions (article) | Khan Academy (19)

      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)

  • Michael Gomez

    8 years agoPosted 8 years ago. Direct link to Michael Gomez's post “In challenge problem 8, I...”

    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?

    • Tomer Gal

      8 years agoPosted 8 years ago. Direct link to Tomer Gal's post “You don't have to know th...”

      Zeros and multiplicity | Polynomial functions (article) | Khan Academy (23)

      You don't have to know this to solve the problem. You can find the correct answer just by thinking about the zeros, and how the graph behaves around them (does it touch the x-axis or cross it). You can click on "I need help!" to see the solution.

      If you want to know how to determine the direction of the graph, check out the next tutorial:

      https://www.khanacademy.org/math/algebra2/polynomial-functions/polynomial-end-behavior/a/end-behavior-of-polynomials

      (10 votes)

  • gingerpotatos

    a year agoPosted a year ago. Direct link to gingerpotatos's post “How do they code the conf...”

    How do they code the confetti when you click the button?

    (9 votes)

  • emmat0802

    a year agoPosted a year ago. Direct link to emmat0802's post “How would I solve f(x)=(2...”

    How would I solve f(x)=(2x-1)(x-5)?

    (4 votes)

    • Kim Seidel

      a year agoPosted a year ago. Direct link to Kim Seidel's post “You have a function with ...”

      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)

  • Anthony

    5 years agoPosted 5 years ago. Direct link to Anthony's post “What if there is a proble...”

    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)

    • kubleeka

      5 years agoPosted 5 years ago. Direct link to kubleeka's post “A polynomial doesn't have...”

      A polynomial doesn't have a multiplicity, only its roots do. The roots of your polynomial are 1 and -2. 1 has multiplicity 3, and -2 has multiplicity 2.

      (5 votes)

  • shub112

    5 years agoPosted 5 years ago. Direct link to shub112's post “Using multiplity how can ...”

    Using multiplity how can you find number of real zeros on a graph

    (3 votes)

    • loumast17

      5 years agoPosted 5 years ago. Direct link to loumast17's post “So first you need the deg...”

      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)

  • sophia.elizabeth.wright

    3 years agoPosted 3 years ago. Direct link to sophia.elizabeth.wright's post “how can you figure out if...”

    how can you figure out if the sign of f on the interval -1 < x < 4 is positive or negative?

    (4 votes)

    • Jerry Nilsson

      3 years agoPosted 3 years ago. Direct link to Jerry Nilsson's post “I guess you're talking ab...”

      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)

Zeros and multiplicity | Polynomial functions (article) | Khan Academy (2024)

FAQs

How to find zeros and multiplicity of a function? ›

Finding Zeros and Their Multiplicities Given a Factored Polynomial. Step 1: Find each zero by setting each factor equal to zero and solving the resulting equation. Step 2: Find the multiplicity of each factor by examining the exponent on the corresponding factor.

What is the zero rule for multiplicity? ›

If the graph crosses the x-axis and appears almost linear at the intercept, it is a single zero. If the graph touches the x-axis and bounces off of the axis, it is a zero with even multiplicity. If the graph crosses the x-axis at a zero, it is a zero with odd multiplicity. The sum of the multiplicities is the degree n.

How do you explain multiplicity? ›

Multiplicity is just the degree (exponent) of each factor, or the amount of times a factor appears. If the multiplicity is even, the graph will “bounce” off the x-axis where that factor is zero. And if the multiplicity is odd, the graph will cross through the x-axis.

Can a polynomial have a repeated zero? ›

The zeros arising from repeated factors of a polynomial function are called repeated zeros . Set to zero, factor and solve. Since appears twice as a factor, appears twice as a zero of the function and is called a repeated zero of the function.

Can 0 have a multiplicity of 2? ›

The number of times a given factor appears in the factored form of the equation of a polynomial is called the multiplicity. The zero associated with this factor, x=2, has multiplicity 2 because the factor (x−2) occurs twice.

How to find zeros of a function? ›

How to Find the Zeroes of a Function. For a linear function, the zero can be found by solving directly. Set the function equal to zero, and then solve for the variable. The zero can also be found by graphing the function.

How to find multiplicities on a graph? ›

In general, if a function ‍ has a zero of odd multiplicity, the graph of y = f ( x ) ‍ will cross the ‍ -axis at that ‍ value. If a function ‍ has a zero of even multiplicity, the graph of y = f ( x ) ‍ will touch the ‍ -axis at that point.

How to find the multiplicity of a root? ›

How do we show that a number c is a root of multiplicity k for a polynomial p? We divide p by (x − c)k and call the resulting ratio g. We substitute c into g and if the claimed multiplicity is right, we get a non-zero number.

How to find multiples of a number? ›

A multiple of a number is a number that is the product of a given number and some other natural number. For example, when we multiply 7 by 3, we get 21, i.e. 7 × 3 = 21. Here, 21 is the multiple of 7.

What are the three types of multiplicity? ›

There are four types of multiplicities: one-to-one, one-to-many, many-to-one, and many-to-many.

What is the multiplicity formula in math? ›

Mathwords: Multiplicity. How many times a particular number is a zero for a given polynomial. For example, in the polynomial function f(x) = (x – 3)4(x – 5)(x – 8)2, the zero 3 has multiplicity 4, 5 has multiplicity 1, and 8 has multiplicity 2.

What is multiplicity in one word? ›

Multiplicity most commonly means “a great variety,” but using the phrase a great variety (or something similar) is much more common than saying multiplicity.

Which polynomial will never have a zero? ›

A constant polynomial p(x) = a ≉ 0 (a polynomial of degree 0) possess no zero .

How many possible zeros can a polynomial have? ›

A polynomial of degree n can have at most n zeros.

Can a polynomial have 2 zeros? ›

Solution: A polynomial is a type of expression in which the exponents of all variables should be a whole number. A polynomial can have any number of zeroes and depends on the degree of the polynomial.

How to determine the multiplicity of a molecule? ›

Find out how many unpaired electrons are there in your molecule. Each unpaired electron has s value = 1/2 ... Spin multiplicity relation (2S+1) will be useful to find the multiplicity of a molecule. You have to arrange the electron properly and find how many unpaired electrons are available (each one has +1/2).

Which function has a zero with a multiplicity of 2? ›

Final answer:

A function has a zero with a multiplicity of 2 if it can be factored such that a term is squared, like f(x) = (x - 3)². Here, the zero (x=3) is repeated twice, thus it has a multiplicity of 2. The multiplicity signifies the number of times the related factor appears in the function.

How to find the turning points of a function? ›

The number of turning points can be found by differentiating the function and setting the derivative equal to zero which will then give the x coordinates of any turning points. The number of solutions found corresponds to the number of turning points.

What is the zero of a polynomial? ›

Zeros of a polynomial can be defined as the points where the polynomial becomes zero as a whole. A polynomial having value zero (0) is called zero polynomial. The degree of a polynomial is the highest power of the variable x. A polynomial of degree 1 is known as a linear polynomial.

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