Lec 37 - Zeroes of Polynomial Functions

Summary

This video script delves into identifying zeros of polynomial functions, emphasizing the factoring technique as a crucial method for finding x-intercepts. It explains how to set polynomial equations to zero, extract common factors, and use quadratic equation-solving skills to simplify the process. The script also highlights the use of technology, such as Desmos, for graphing and verifying results, providing step-by-step examples for quadratic, cubic, and higher-degree polynomials, and discusses strategies for x and y-intercepts.

Takeaways

• 🔢 The zeros of a polynomial function are the values of x for which the polynomial equals zero.
• 📉 Identifying zeros of quadratic functions can involve graphing the function and finding the x-intercepts.
• ➗ Factoring the polynomial is a key method to find the zeros of polynomial functions.
• 🔍 Factoring can involve looking for the greatest common factor or using techniques like grouping and trinomial factoring.
• 📊 Graphical tools such as Desmos can help visualize and verify the zeros of polynomial functions.
• 📐 For higher-degree polynomials, finding zeros can be challenging and may require trial and error or advanced techniques.
• 📝 Setting the polynomial function equal to zero and factoring can reveal the zeros of the function.
• 🧮 Example: For the polynomial x^6 - 8x^4 + 16x^2, factoring out the greatest common factor x^2 and further factoring the quadratic term reveals the zeros at x = 0, 2, and -2.
• 📈 Verification using technology can confirm the identified zeros, as shown in the provided examples.
• 🌐 Understanding and applying these techniques are crucial for solving polynomial functions and finding their zeros effectively.

Q & A

What is a zero of a polynomial function?

A zero of a polynomial function is a value of x for which the function f(x) is equal to 0.

How can the zeros of quadratic functions be identified?

Zeros of quadratic functions can be identified by graphing the function, using the axis of symmetry, factoring the quadratic into intercept form, or setting the quadratic function to zero and solving for x.

What is an important technique for finding the zeros of polynomial functions?

Factoring the polynomial function is an important technique for finding the zeros.

What should you do if you cannot factor a polynomial easily?

If you cannot factor a polynomial easily, you can try using random values to find zeros and then use synthetic division to divide the polynomial by the corresponding factor.

What is the general approach for finding zeros of higher-degree polynomials?

For higher-degree polynomials, you can try trial and error methods, use knowledge of lower-degree polynomials, and graphing tools to identify zeros.

What is the first step in finding the x-intercepts of a polynomial function by factoring?

The first step is to set the polynomial function f(x) equal to 0.

How can you verify the zeros of a polynomial function?

You can verify the zeros of a polynomial function using graphical tools such as Desmos.

What is an example of factoring a polynomial to find its zeros?

For the polynomial x^6 - 8x^4 + 16x^2, you can factor out the greatest common factor x^2, then factor the remaining polynomial as a quadratic, and solve for x.

How can you identify x-intercepts for a polynomial in factored form?

For a polynomial in factored form, you can set each factor to zero and solve for x to find the x-intercepts.

What is the y-intercept of a polynomial function, and how is it found?

The y-intercept of a polynomial function is the value of the function when x is 0. It is found by substituting x = 0 into the polynomial and solving for the function value.

Outlines

00:00

📚 Understanding Zeros of Polynomial Functions

This paragraph introduces the concept of zeros in polynomial functions, where zeros are the x-values that make the function equal to zero. It recalls the methods used for finding zeros of quadratic functions, such as graphing and factoring. The speaker emphasizes the importance of identifying the axis of symmetry and using it to find zeros by observing where the graph intersects the x-axis. The paragraph also discusses the trial and error method and the use of random values to guess potential factors, which can then be verified by polynomial division. It concludes by noting the complexity of finding zeros in higher degree polynomials and the lack of straightforward formulas beyond quadratic equations.

05:03

🔍 Factoring Techniques for Polynomial Zeros

The second paragraph delves into various factoring techniques to find zeros of polynomials. It suggests looking for the greatest common factor and using factor by grouping as manageable approaches. For higher degree polynomials, trinomial factoring is recommended. The paragraph also mentions the use of graphical tools like Desmos to determine intercepts by graphing the function. The speaker illustrates the process with an example, showing how to factor out common terms, apply quadratic factoring to the remaining expression, and equate each factor to zero to find the x-intercepts.

10:06

📉 Graphical Verification of Polynomial Zeros

This paragraph focuses on the verification of polynomial zeros using graphical tools. It provides a step-by-step example of how to find x-intercepts for a given polynomial by setting the function equal to zero, identifying common factors, and factoring the expression. The speaker then demonstrates solving for x-intercepts by factoring a cubic polynomial and using the quadratic formula. The paragraph concludes with the use of technology to verify the roots graphically, ensuring the accuracy of the identified x-intercepts.

15:14

📈 Identifying Intercepts in Factored Polynomial Forms

The final paragraph discusses identifying x-intercepts and y-intercepts of polynomial functions given in factored form. It explains that x-intercepts can be visually guessed and mathematically confirmed by setting the function equal to zero. For y-intercepts, the process involves substituting x with zero in the polynomial expression. The paragraph provides an example of finding both x and y-intercepts for a specific polynomial and uses graphing technology to verify the results, showcasing the function's graph with identified intercepts.

Keywords

💡Zeros of Polynomial Functions

Zeros of a polynomial function are the values of x for which the function f(x) equals zero. These are crucial in understanding the points where the graph of the polynomial intersects the x-axis. For instance, the script explains how finding zeros is essential for solving polynomial equations and is demonstrated with both quadratic and cubic functions.

💡Quadratic Functions

Quadratic functions are polynomial functions of degree 2, typically in the form f(x) = ax^2 + bx + c. The script discusses methods like graphing and factoring to find their zeros. Identifying the axis of symmetry and plotting ordered pairs are mentioned as techniques to solve these functions.

💡Factoring

Factoring involves expressing a polynomial as a product of its factors. It's a fundamental technique for finding the zeros of polynomial functions, as explained in the script. Factoring helps simplify the polynomial, making it easier to solve. The example of factoring x^6 - 8x^4 + 16x^2 demonstrates its importance.

💡Graphing

Graphing is the process of plotting a polynomial function on a graph to visually identify its zeros. The script mentions using graphing as a method to find the x-intercepts of quadratic and higher-order polynomials. Tools like Desmos are suggested for graphing polynomials accurately.

💡Common Monomial

A common monomial is the greatest common factor that can be factored out from each term in a polynomial. Identifying this simplifies the polynomial, making further factoring easier. The script illustrates this with the example of factoring out x^2 from x^6 - 8x^4 + 16x^2.

💡Binomial Factoring

Binomial factoring is a technique where pairs of terms in a polynomial are grouped and factored. This method is useful for polynomials with four or more terms. The script shows this with the polynomial x^3 - 4x^2 - 3x + 12, where binomial pairs are factored to simplify the equation.

💡Trinomial Factoring

Trinomial factoring involves factoring polynomials with three terms. This is a common method for solving quadratic equations and higher-degree polynomials. The script refers to this technique when discussing how to handle complex polynomials by breaking them down into manageable trinomials.

💡x-intercept

The x-intercept of a polynomial function is the point where the graph crosses the x-axis, indicating the zeros of the function. Finding x-intercepts is a primary goal when solving polynomial equations. The script provides detailed steps and examples to determine x-intercepts through factoring and graphing.

💡Quadratic Formula

The quadratic formula, x = (-b ± √(b^2 - 4ac)) / 2a, is used to find the zeros of quadratic functions. The script mentions it as a reliable method for solving quadratic equations when factoring is not feasible. It contrasts the quadratic formula's simplicity with the complexity of higher-degree polynomial solutions.

💡Desmos

Desmos is an online graphing tool used to visualize polynomial functions and find their zeros. The script highlights Desmos as a valuable resource for verifying the solutions of polynomial equations by graphing the functions and identifying x-intercepts accurately.

💡y-intercept

The y-intercept of a polynomial function is the point where the graph crosses the y-axis, indicating the value of the function when x = 0. The script explains how to find the y-intercept by substituting x = 0 into the polynomial and calculating the result, as shown in the example with g(x).

Highlights

Zeros of a polynomial function are the values of x for which the polynomial equals zero.

Identifying zeros of quadratic functions can be done by graphing, factoring, or using the quadratic formula.

Factoring is a crucial technique for finding zeros of polynomial functions.

For higher degree polynomials, trial and error and knowledge of lower degree polynomials are often used.

Graphical tools like Desmos can be used to determine intercepts of polynomial functions.

Greatest common factor and factoring by grouping are effective methods for factoring polynomials.

Trinomial factoring is useful for high-degree polynomials.

Setting the polynomial function equal to zero and solving for x gives the x-intercepts.

Verification of x-intercepts can be done using graphing tools.

Example: Finding x-intercepts of a polynomial function by factoring and using Desmos to verify.

Factoring quadratic-like expressions within higher degree polynomials can simplify finding zeros.

Example: Using factoring techniques to find x-intercepts of a cubic polynomial.

Identifying common monomials and binomials is key in factoring polynomial expressions.

Example: Finding x and y-intercepts of a polynomial given in factored form.

Setting g(x) to zero and solving gives x-intercepts; substituting x=0 gives y-intercepts.

Transcripts

So, let us focus on Zeros of Polynomial Functions. So, for clarity, let us recall what is zero

of a polynomial function. If f is a polynomial function, then the values of x for which f

x is equal to 0 is called zero of f. A value of x for which f x is equal to 0 is called

zero of f . Now, when we studied quadratic functions,

we we had several methods of identifying the zeros of the quadratic functions. For example,

we actually tried to graph the quadratic function because we knew some techniques, we actually

plotted set of ordered pairs on a graph paper and join the curve smoothly, then we identified

it is crucial to identify axis of symmetry and around axis of symmetry you can plot and

wherever it intersects x axis, we will call that as a zero of a function. This is how

we identified quadratic zeros of quadratic functions.

Another way that we used which will be helpful here is factoring the quadratic function into

factors given a quadratic function identify the factors and write thepolynomial into intercept

form . If you are able to do that, then you have

again identified zeros of the polynomial because when you said that quadratic function to be

equal to zero and if it is in a factored form, all the coefficients corresponding to that

factor will be all the numbers corresponding to that factor will be zeros of the polynomial

function . So, now, we will focus on the factoring component

of polynomial functions. So, if the equation of the polynomial function can be factored,

then we can set that each factor to be equal to 0 and solve for zeros. This is an important

step. But it as we have seen in quadratic functions, this is not always possible .

In such case, if you put some random values, if you throw in some random values in the

function and you get something like x is equal to a, you will get the value to be 0 that

is also helpful. Then, you can guess that x minus a is a factor and you can use the

previous video to divide the polynomial by x minus a which will give you the remainder

term. And that remainder, you can actually figure

out whether you can ah; you can consider factoring for that remainder or not all these things

are possible or the other factor it is not remainder sorry it is the other factor . So,

these are some possible ways hm . Up to quadratic equations, we had some easy

ways out easy way out like given the equation of a quadratic function, we can use this method

to find x intercept because x for x intercepts, we get zeros that is what I explained earlier

also. So,you can find x intercepts and you will easily get this. You can use the similar

technique of finding x intercepts for a general polynomial function also , but it is very

difficult to plot a polynomial function ok. Given a graph of a polynomial function and

you have identified based on our previous criteria, you have identified that this is

a polynomial function, you can guess what are the zeros of the polynomial function that

way this this this statement helps. But, if you go for higher order polynomials

that is general polynomials, this can become messy, it can be really challenging. Quadratic

equations can be easily solved using quadratic formula we have a solution for quadratic equations.

But, the cubic and four-degree polynomials have some formulae which you may study in

your tutorials, but they are not easy enough to remember.

And, for higher degree polynomials, you do not have any idea of how to approach finding

zeros of the polynomial functions you have to go by trial and error method and whatever

knowledge you have about square ,quadratic, linear and cubic polynomials .

So, let us summarize what we have; what we have discussed just now. If I want to identify

zeros of polynomial functions, the factoring technique is a crucial technique . So, what

you can do is you can look at the polynomial and if you look at the polynomial, there is

one easy way out that if you can identify the greatest common factor that is the greatest

monomial that can be taken out common you can use that technique.

Once , if there is no such technique, if once that is available, the polynomial ismore or

less manageable, then you can use the technique of factor by grouping. So, you can create

groups in that and see whether a anything is coming out common that is another technique.

Another thing is you can instead of handling groups, you can decide to handle three terms

at a time so, that is a trinomial factoring. This will be helpful when you have very high

degree polynomial. So, these are the common methods for factoring the polynomials. Once

you can factor the polynomials each of them can be equated to 0 by writing a polynomial

in a factored form ok . And then finally, if you are not very sure,

then you can use some graphical tools which are available these days on computer or on

the netone such tool is Desmos which we are using in our presentations .

So, you can use those tools to determine the intercepts. In these tools basically, you

will give of equation of a function and it will be graphed they will give the they will

project the graph of a function right. So, this is our zeros of the polynomials and factoring

play a crucial role. To understand this , let us see how to find

x-intercept of a polynomial function by factoring . So, what we have discussed just now is we

have set the equation that is f of x is equal to 0 in order tofacilitate factoring f of

x equal to 0, then if the if the polynomial is given in factor form; factored form then

equate each of them to be equal to 0 which we have seen for quadratic case also.

If it is not given in factor factored form, first in that you will look for is you take

out some common monomial that is available in all the terms if that is that is there

and you have taken out or if that is not there still you can go to the second step that is

whatever at the rest of the terms you can factor them into factorable binomials or trinomials

you look for try to look for combinations which we have done successfully for quadratic

equations well doing the factoring. So, you can do a similar thing over here .

And then finally, set each factor equal to zero that will give you the x-intercept . This

is the; this is the strategy that we will follow for finding x-intercept of polynomial

function by the method of factoring. So, let us look at this example where we will

follow the steps of the algorithm. So, the a a question says find x-intercepts of a function

x raised to 6 minus 8x raised to 4 plus 16x square.

So, as per our algorithm or as per the steps given in the previous slide, I will set f

x to be equal to 0 that is x raised to 6 minus 8x raised to 4 plus 16x square is equal to

0. Now, you look at greatest common factor, a monomial that is common in all these terms

that is x square. So, what I will do is I will separate out this x square, I have taken

out this x square and now, you look at the other factor that is x raised to 4 minus 8x

square plus 16. Now, this factor can be related to our quadratic

equation of the form t square minus 8t plus 16. Can I factor this quadratic equation because

there is no term corresponding to x raised to 1 and there is no there are no odd terms

essentially . So, I can use this and I can leverage the

skill of quadratic equations to solve this equation and from quadratic equation point

of view, I know this is t minus 4 the whole square equal to 0. So, instead of t here,

it is x square. So, that will give me x square into x square minus 4 the whole square is

equal to 0 . Now everything is looks in the form of x square.

So, what are the values of x? What are the feasible values of x? Those will be the x-intercepts.

So, you can put x square is so, this this will give me x square is equal to 0 or x square

minus 4 is equal to 0 ok . So, x square minus 4 can further be factored into x minus 2 x

plus 2 equal to 0 and with this understanding, I can write x is equal to 0, 2 and minus 2

are the intercepts of f x-intercepts of f ok.

Now,as per the last step in the algorithm, you want to verify this result. How will you

verify this result? Using the technology . So, using Desmos, I have drawn this graph and

you can verify that x is equal to minus 2 which is here, x is equal to 0 which is here

and x is equal to 2 which is here are all x-intercepts of a polynomial function given

by these f x ok. So, this is how we will identify x-intercepts. Let let us understand this strategy

by looking at one more example . So, now here, we have been asked to find x-intercept

of a polynomial function which is a cubic polynomial function x cube minus 4x square

minus 3x plus 12 fine. So, as per our set up, this first step is set f x is equal to

0. So, you have set f x is equal to 0 that that is essentially gives me x cube minus

4x square minus 3x plus 12 is equal to 0. Then, the second a step if you have any common

monomial, there is no common monomial because the last term is a constant term so, you cannot

figure out a common monomial. Then is there any pattern? Can you look at two-two terms

each binomials or trinomials because there are four terms , it is better to look at binomial

terms . So, if you look at the first two terms , you

can see that you can throw out x square as a common thing , if you throw out x square

as a common thing, then you will be stayed with x minus 4 as a term as a one factor . And

if you look at these two terms, then again if you take out 3 common minus 3 common, then

you will get x minus 4. So, using the technique of binomial binomials

in this case, I am able to see this kind of factoring possible . Good, that essentially

means I can rewrite this expression as x square minus 3 into x minus 4 equal to 0 .

Then, I want to solve this x square minus 3 that is all is remaining which is a quadratic

equation. So, you can easily solve using quadratic formula or a factoring, but here in this case,

I know the factors so, that will be x minus root 3 and x plus root 3 equal to 0.

And therefore; therefore, the solution of this quadratic equation is well known that

is x is equal to 4 plus root 3 and minus root 3 are the x-intercepts of the function. The

final step is I want to verify using some technology or a graphing tool. This is the

graph of a function . So, in this case, you can easily verify there

are three roots: first root this one which is a occurs it occurs at minus root 3 , this

one this is plus root 3, this one is 4 . So, these are the fourthese are the three roots

of a cubic polynomial . Roots or x-intercepts or a zeros of a cubic polynomial ok.

So, let us go ahead and see let me remove this blocks. Another example where I want,

I am interested in finding x-intercepts as well as y-intercepts of a polynomial function

which is given in a factored form . So, the polynomial is given in factored form.

So, visually you will be able toguess the roots. So, as a standard set up, we will set

g x to be equal to 0 . Once you said g x to be equal to 0, it is very clear that x is

equal to 1 and x is equal to minus 3 are the x-intercepts of f. What about y-intercept?

What is the y-intercept at all ? So, y-intercept is where x is given to be 0. So, simply substitute

x to be equal to 0 in the expression of g x you will get g of 0 which is 0 into minus

1 the whole square that is 1 plus 0 plus 3 that is 3 so, 1 into 3 is 3 so, your g 0 is

3. So, this is how you will figure out x-intercepts

and y-intercepts of the function and this is the graph of that function . Using technology,

I have verify.