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We learned what a Polynomial is here in the Introduction to Multiplying Polynomials section. As a review, here are some polynomials, their names, and their degrees. Remember that the degree of the polynomial is the highest exponent of one of the terms (add exponents if there are more than one variable in that term). We learned that a Quadratic Function is a special type of polynomial with degree 2; these have either a cup-up or cup-down shape, depending on whether the leading term (one with the biggest exponent) is positive or negative, respectively. There are certain rules for sketching polynomial functions, like we had for graphing rational functions.

Again, the degree of a polynomial is the highest exponent if you look at all the terms (you may have to add exponents, if you have a factored form). The leading coefficient of the polynomial is the number before the variable that has the highest exponent (the highest degree). So for, the degree is 4, and the leading coefficient is 5; for, the degree is 7 (add exponents since the polynomial isn’t multiplied out and don’t forget the x to the first power), and the leading coefficient is –10 (you can tell by the –5 in front and the 2x in the factor with the highest exponent). Each factor in a polynomial has what we call a multiplicity, which just means how many times it’s multiplied by itself in the polynomial (its exponent). Remember that x – 4 is a factor, while 4 is a root (zero, solution, x-intercept, or value). Now we can use the multiplicity of each factor to know what happens to the graph for that root – it tells us the shape of the graph at that root. Also note that you won’t be able to determine how low and high the curves are when you sketch the graph; you’ll just want to get the basic shape. Here are a few problems where we use the Conjugate Zeroes Theorem and Complex Conjugate Zeroes Theorem (also called Conjugate Root Theorem or Conjugate Pair Theorem), which states that if is a root, then so is . The complex form of this theorem states that if is a root, then so is . Also remember that when we factor to solve quadratics or any polynomials, we can never just divide by factors (with variables) on both sides to get rid of them. If we do this, we may be missing solutions!

Many times we’re given a polynomial in Standard Form, and we need to find the zeros or roots. For higher level polynomials, the factoring can be a bit trickier, but it can be sort of fun — like a puzzle! Remember that if we divide a polynomial by “x – c” and get a remainder of 0, then “x – c” is a factor of the polynomial and “c” is a root.

When we want to factor and get the roots of a higher degree polynomial using synthetic division, it can be difficult to know where to start! In the examples so far, we’ve had a root to start, and then gone from there. For a polynomial function with integers as coefficients (no fractions or decimals), if p = the factors of the constant (in our case, d), and q = the factors of the highest degree coefficient (in our case, a), then the possible rational zeros or roots are where p are all the factors of d above, and q are all the factors of a above. Remember that factors are numbers that divide perfectly into the larger number; for example, the factors of 12 are 1, 2, 3, 4, 6, and 12.

The rational root test help us find initial roots to test with synthetic division, or even by evaluating the polynomial to see if we get 0. However, it doesn’t make a lot of sense to use this test unless there are just a few to try, like in the first case above. Now let’s try to find roots of polynomial functions without having a first root to try. Also remember that you may end up with imaginary numbers as roots, like we did with quadratics.

Take out any Greatest Common Factors (GCFs) of the polynomial, and you’ll have to set those to 0 too, to get any extra roots. For example, if you take an x out, you’ll add a root of “0”. If you have access to a graphing calculator, graph the function and determine if there are any rational zeros with which you can use synthetic division. If you don’t have a calculator, guess a possible rational zero using the method above.

There are a couple of theorems that you’ll learn about that will help you evaluate polynomials (for a given x, find the y) and also be able to quickly tell if a given number is a root. There’s another really neat trick out there that you may not talk about in High School, but it’s good to talk about and pretty easy to understand. Yes, and it was named after a French guy! The DesCartes’ Rule of Signs will tell you the number of positive and negative real roots of a polynomial by looking at the sign changes of the terms of that polynomial.

We talked a little bit about the Complex Conjugate Zeros Theorem here when we talked about all the steps required to find all the factors and roots of a polynomial.

A polynomial function is a relation between some variable to another variable with a few restrictions to limit its behavior.

In this tutorial, we will define a polynomial function and learn how to graph them and study the shape of the graph.

The following are examples of true polynomial functions since they follow all the above restrictions. The following do not follow the above restrictions and therefore cannot be considered polynomial functions. The degree of a polynomial function is the highest exponential value of the independent variable in the function.

There are a few defining features that all polynomial functions have in common that are extremely useful to note when graphing. The list seems long, but when handled one at a time systematically, it isn't too much to worry about.

To actually graph a polynomial function, it is best to find all the defining features defined above. If the degree of the polynomial is 1, like in the equation $y = 3x+2$, then the graph is a line. The end behavior of a polynomial function is a description of how the polynomial behaves as it approaches positive and negative infinity.

There are four possible end behaviors that could occur assuming the degree of the function is not 0.

This shape results if the leading term is positive and the degree of the polynomial is even. This shape results if the leading term is positive and the degree of the polynomial is odd. This shape results if the leading term is negative and the degree of the polynomial is even.

This shape results if the leading term is negative and the degree of the polynomial is odd. So, let us take the following polynomial function and use it as an example to determine its end behavior.

The x-intercepts of a polynomial are where the polynomial intersects the x-axis on the real coordinate plane. Sometimes, the polynomial is already factored for you, and so it is easy to identify the x-intercepts. We could commence the normal procedure for finding x-intercepts by setting the y equal to zero and solving. The graph touches and rebounds the x-axis when the particular x-intercept has an even multiplicity, like 2, 4, 6, etc.

If the intercept has an even multiplicity, meaning it occurs twice, four times, eight times, etc., then the graph appears to touch the x-axis and then bounces off in the same direction it came from. If the x-intercept has an odd multiplicity, meaning it occurs three times, five times, etc., then the graph kind of lingers around the interception point before passing through. The y-intercept of a polynomial is where the graph crosses the y-axis on the real coordinate plane. The derivative is more of a calculus topic, but nonetheless it is extraordinarily helpful when extracting information from a polynomial in order to graph it. The extrema are the minimum points and maximum points within the middle behavior of the polynomial. So, the degree of a polynomial tells the most number of possible x-intercepts the function can have. It is impossible to get a real number result when taking the square root of a negative number. It is possible, however, to end up trying to graph a polynomial function that has one x-intercept and no extrema, or no x-intercepts and one extrema.

Since it is in factored form and not in standard form, we cannot identify things like y-intercept or end behavior easily. When the slope of the polynomial is zero, an extrema occurs, so we will solve the derivative when it is set equal to 0. The factor in the parentheses looks a bit daunting to factor, and there is of course the possibility that it will not factor. The only way to get better at graphing polynomials is by actually graphing a bunch of them. Notice that the negative part of the graph is more of a “cup down” and the positive is more of a “cup up”. If there is no exponent for that factor, the multiplicity is 1 (which is actually its exponent!) And remember that if you sum up all the multiplicities of the polynomial, you will get the degree!

This seems very limiting, but there are still so many possible behaviors polynomials functions can exhibit. Sometimes, it is written in descending degree order, and sometimes it is a representation of a bunch of factors. This form directly exposes the prime factors of a particular polynomial and therefore makes it very easy to determine x-intercepts of the graph.

What this means is that the degree is the largest exponent that can be found on the independent variable1 when the polynomial is simplified (all parentheses gone and such).

This polynomial function is not simplified, so we first simplify it to $y=x^2-x-2$ so that the degree becomes obvious, 2.

These things since they are common for all polynomials are used as the base for graphing the polynomial in an easy and quick fashion.

This gives a bunch of information about the shape and positioning of the polynomial which makes it possible to sketch a graph. If it has a degree of 2, like in $y = x^2+3x-2$, then the shape is a parabola, which is like a U. The end behavior is determined by the sign of the coefficient of the leading term (is it positive or negative) and the degree. There is no easy way to do this except by factoring, and in most cases the polynomial won't factor properly3. Depending on the multiplicity of an x-intercept, you obtain one of the three types of x-intercepts from the first paragraph in this section. The graph does actually pass through, but it is sort of delayed before actually passing through, like in the image. Unlike with x-intercepts, there can only be one y-intercept, and there always is one y-intercept. This is a contrast from x-intercepts where y was equal to 0; this time the other variable is 0.

The derivative is a secondary function that exposes the slope of the original polynomial at given points. There is a complex calculus formula that defines the derivative of a function, but our concern is simply graphing a polynomial function.

To find its derivative, we apply the formula to every single term and add them together Ignore the constant term.

The next section will explain the importance of the derivative and how it can be used to graph the original polynomial. And also, higher degree polynomials are much harder to find extrema for since it requires the solving of higher degree derivatives.

The above information is all that is needed to really sketch an accurate drawing of the function. Write out little arrows indicating where the graph will enter from and where it will leave.

Plot the points, note multiplicities, note end behaviors, and simply connect the points together. Find out what kind of problem it is so you can better prepare your strategy for graphing it. If you look at the y-intercept and extrema, there are very large y-values, such as the -36 and -69.63. Here are some practice problems to try out on your own so you can use the information in this tutorial firsthand. There is a ton more to derivative than that, but it is a calculus idea, and we will only discuss this particular aspect. Polynomial functions are special in that they contain no discontinuities in their behavior, have distinct slopes and features, and have end behaviors that approach infinity. There are two important forms of polynomials that are extremely useful when extracting information out of them. So, for the polynomial $y=x^2+3x-2$, the degree is 2 since that is the largest exponential value.

The first thing it tells us is the general shape of the polynomial, such as does it start from the bottom and continue to the top, or does it start at the top and curve its way back to the top? All odd degree functions have unlimited range, which means they seem to go on forever in both up and down directions. It occurs once, so it passes through and continues along with the normal path that it takes. This line is said to have a slope, a sort of definition to how steep it is compared to the x-axis. When the slope is zero, this means that the point is either the top of a hill or bottom of a valley, the derivative is crossing the x-axis. The extrema are unfortunately not usually located on either of the axes; they are usually some point in one of the four quadrants. As a matter of definition, an extrema mathematically occurs when the slope of the polynomial at a given x location is 0. The power algorithm yields a derivative of $f'(x) =3x^2+7$, and when we solve for 0, then we find there are no real solutions for the derivative.

If a certain x-intercept has a multiplicity other than one, write out the multiplicity beside the point.

When a problem is given in any form other than standard, it is always best to turn it into that form.

This will leave you with three y values which you can pair up with their parent x value to obtain the extrema.

If you are graphing multiple equations on the same graph, it is best to color code the graphs or label them. These functions are excellent for demonstrating real life situations, such as banking trends. The polynomial $y = (x+6)(x-4)+5x^2$ is not factored because a 5x2 is being added rather than multiplied. Whenever you first begin observing a polynomial in preparation to graph, always examine the degree first.

Even degree functions have either an absolute minimum or maximum, so they don't go on forever in both directions, just one.

The slope is 3 everywhere since the value of y increases by three for every unit increase of x.

The job of the derivative is to tell us specifically the slope of every point on the curve4.

An imaginary root cannot be graphed as an x-intercept, and they do not provide much visual clue when the polynomial is graphed. The factored form allows us to set every factor to zero therefore making the entire thing possible to solve. To find the derivative, we use the power rule, $ax^n \rightarrow anx^{n-1}$ on each term when the polynomial is in standard form.

We therefore must establish a different viewing window, or in other words, different dimensions. The end behavior is totally dependent on the leading term of the polynomial function when simplified2. Therefore, there is a very large graphical connection between the function and its derivative.

Once you find the x part, simply substitute the values into the original polynomial, and the respective y parts will emerge. Nonetheless, they are still an important part of the function, and it is best to identify the imaginary roots whenever possible. It is a little more advanced than quadratic FOIL, but it is very much the same concept; every term in a factor gets multiplied by every term in another. An odd degree polynomial can have no extrema, but an even degree function must have at least one.

That doesn't mean there will be five x-intercepts; it just means there can be at most five x-intercepts, so expect that. Mark these down for later, and do not forget to indicate that the intercept (-3,0) has a multiplicity!

It is fully factored since there are no other factors that can be further simplified or factored.

Again, the degree of a polynomial is the highest exponent if you look at all the terms (you may have to add exponents, if you have a factored form). The leading coefficient of the polynomial is the number before the variable that has the highest exponent (the highest degree). So for, the degree is 4, and the leading coefficient is 5; for, the degree is 7 (add exponents since the polynomial isn’t multiplied out and don’t forget the x to the first power), and the leading coefficient is –10 (you can tell by the –5 in front and the 2x in the factor with the highest exponent). Each factor in a polynomial has what we call a multiplicity, which just means how many times it’s multiplied by itself in the polynomial (its exponent). Remember that x – 4 is a factor, while 4 is a root (zero, solution, x-intercept, or value). Now we can use the multiplicity of each factor to know what happens to the graph for that root – it tells us the shape of the graph at that root. Also note that you won’t be able to determine how low and high the curves are when you sketch the graph; you’ll just want to get the basic shape. Here are a few problems where we use the Conjugate Zeroes Theorem and Complex Conjugate Zeroes Theorem (also called Conjugate Root Theorem or Conjugate Pair Theorem), which states that if is a root, then so is . The complex form of this theorem states that if is a root, then so is . Also remember that when we factor to solve quadratics or any polynomials, we can never just divide by factors (with variables) on both sides to get rid of them. If we do this, we may be missing solutions!

Many times we’re given a polynomial in Standard Form, and we need to find the zeros or roots. For higher level polynomials, the factoring can be a bit trickier, but it can be sort of fun — like a puzzle! Remember that if we divide a polynomial by “x – c” and get a remainder of 0, then “x – c” is a factor of the polynomial and “c” is a root.

When we want to factor and get the roots of a higher degree polynomial using synthetic division, it can be difficult to know where to start! In the examples so far, we’ve had a root to start, and then gone from there. For a polynomial function with integers as coefficients (no fractions or decimals), if p = the factors of the constant (in our case, d), and q = the factors of the highest degree coefficient (in our case, a), then the possible rational zeros or roots are where p are all the factors of d above, and q are all the factors of a above. Remember that factors are numbers that divide perfectly into the larger number; for example, the factors of 12 are 1, 2, 3, 4, 6, and 12.

The rational root test help us find initial roots to test with synthetic division, or even by evaluating the polynomial to see if we get 0. However, it doesn’t make a lot of sense to use this test unless there are just a few to try, like in the first case above. Now let’s try to find roots of polynomial functions without having a first root to try. Also remember that you may end up with imaginary numbers as roots, like we did with quadratics.

Take out any Greatest Common Factors (GCFs) of the polynomial, and you’ll have to set those to 0 too, to get any extra roots. For example, if you take an x out, you’ll add a root of “0”. If you have access to a graphing calculator, graph the function and determine if there are any rational zeros with which you can use synthetic division. If you don’t have a calculator, guess a possible rational zero using the method above.

There are a couple of theorems that you’ll learn about that will help you evaluate polynomials (for a given x, find the y) and also be able to quickly tell if a given number is a root. There’s another really neat trick out there that you may not talk about in High School, but it’s good to talk about and pretty easy to understand. Yes, and it was named after a French guy! The DesCartes’ Rule of Signs will tell you the number of positive and negative real roots of a polynomial by looking at the sign changes of the terms of that polynomial.

We talked a little bit about the Complex Conjugate Zeros Theorem here when we talked about all the steps required to find all the factors and roots of a polynomial.

A polynomial function is a relation between some variable to another variable with a few restrictions to limit its behavior.

In this tutorial, we will define a polynomial function and learn how to graph them and study the shape of the graph.

The following are examples of true polynomial functions since they follow all the above restrictions. The following do not follow the above restrictions and therefore cannot be considered polynomial functions. The degree of a polynomial function is the highest exponential value of the independent variable in the function.

There are a few defining features that all polynomial functions have in common that are extremely useful to note when graphing. The list seems long, but when handled one at a time systematically, it isn't too much to worry about.

To actually graph a polynomial function, it is best to find all the defining features defined above. If the degree of the polynomial is 1, like in the equation $y = 3x+2$, then the graph is a line. The end behavior of a polynomial function is a description of how the polynomial behaves as it approaches positive and negative infinity.

There are four possible end behaviors that could occur assuming the degree of the function is not 0.

This shape results if the leading term is positive and the degree of the polynomial is even. This shape results if the leading term is positive and the degree of the polynomial is odd. This shape results if the leading term is negative and the degree of the polynomial is even.

This shape results if the leading term is negative and the degree of the polynomial is odd. So, let us take the following polynomial function and use it as an example to determine its end behavior.

The x-intercepts of a polynomial are where the polynomial intersects the x-axis on the real coordinate plane. Sometimes, the polynomial is already factored for you, and so it is easy to identify the x-intercepts. We could commence the normal procedure for finding x-intercepts by setting the y equal to zero and solving. The graph touches and rebounds the x-axis when the particular x-intercept has an even multiplicity, like 2, 4, 6, etc.

If the intercept has an even multiplicity, meaning it occurs twice, four times, eight times, etc., then the graph appears to touch the x-axis and then bounces off in the same direction it came from. If the x-intercept has an odd multiplicity, meaning it occurs three times, five times, etc., then the graph kind of lingers around the interception point before passing through. The y-intercept of a polynomial is where the graph crosses the y-axis on the real coordinate plane. The derivative is more of a calculus topic, but nonetheless it is extraordinarily helpful when extracting information from a polynomial in order to graph it. The extrema are the minimum points and maximum points within the middle behavior of the polynomial. So, the degree of a polynomial tells the most number of possible x-intercepts the function can have. It is impossible to get a real number result when taking the square root of a negative number. It is possible, however, to end up trying to graph a polynomial function that has one x-intercept and no extrema, or no x-intercepts and one extrema.

Since it is in factored form and not in standard form, we cannot identify things like y-intercept or end behavior easily. When the slope of the polynomial is zero, an extrema occurs, so we will solve the derivative when it is set equal to 0. The factor in the parentheses looks a bit daunting to factor, and there is of course the possibility that it will not factor. The only way to get better at graphing polynomials is by actually graphing a bunch of them. Notice that the negative part of the graph is more of a “cup down” and the positive is more of a “cup up”. If there is no exponent for that factor, the multiplicity is 1 (which is actually its exponent!) And remember that if you sum up all the multiplicities of the polynomial, you will get the degree!

This seems very limiting, but there are still so many possible behaviors polynomials functions can exhibit. Sometimes, it is written in descending degree order, and sometimes it is a representation of a bunch of factors. This form directly exposes the prime factors of a particular polynomial and therefore makes it very easy to determine x-intercepts of the graph.

What this means is that the degree is the largest exponent that can be found on the independent variable1 when the polynomial is simplified (all parentheses gone and such).

This polynomial function is not simplified, so we first simplify it to $y=x^2-x-2$ so that the degree becomes obvious, 2.

These things since they are common for all polynomials are used as the base for graphing the polynomial in an easy and quick fashion.

This gives a bunch of information about the shape and positioning of the polynomial which makes it possible to sketch a graph. If it has a degree of 2, like in $y = x^2+3x-2$, then the shape is a parabola, which is like a U. The end behavior is determined by the sign of the coefficient of the leading term (is it positive or negative) and the degree. There is no easy way to do this except by factoring, and in most cases the polynomial won't factor properly3. Depending on the multiplicity of an x-intercept, you obtain one of the three types of x-intercepts from the first paragraph in this section. The graph does actually pass through, but it is sort of delayed before actually passing through, like in the image. Unlike with x-intercepts, there can only be one y-intercept, and there always is one y-intercept. This is a contrast from x-intercepts where y was equal to 0; this time the other variable is 0.

The derivative is a secondary function that exposes the slope of the original polynomial at given points. There is a complex calculus formula that defines the derivative of a function, but our concern is simply graphing a polynomial function.

To find its derivative, we apply the formula to every single term and add them together Ignore the constant term.

The next section will explain the importance of the derivative and how it can be used to graph the original polynomial. And also, higher degree polynomials are much harder to find extrema for since it requires the solving of higher degree derivatives.

The above information is all that is needed to really sketch an accurate drawing of the function. Write out little arrows indicating where the graph will enter from and where it will leave.

Plot the points, note multiplicities, note end behaviors, and simply connect the points together. Find out what kind of problem it is so you can better prepare your strategy for graphing it. If you look at the y-intercept and extrema, there are very large y-values, such as the -36 and -69.63. Here are some practice problems to try out on your own so you can use the information in this tutorial firsthand. There is a ton more to derivative than that, but it is a calculus idea, and we will only discuss this particular aspect. Polynomial functions are special in that they contain no discontinuities in their behavior, have distinct slopes and features, and have end behaviors that approach infinity. There are two important forms of polynomials that are extremely useful when extracting information out of them. So, for the polynomial $y=x^2+3x-2$, the degree is 2 since that is the largest exponential value.

The first thing it tells us is the general shape of the polynomial, such as does it start from the bottom and continue to the top, or does it start at the top and curve its way back to the top? All odd degree functions have unlimited range, which means they seem to go on forever in both up and down directions. It occurs once, so it passes through and continues along with the normal path that it takes. This line is said to have a slope, a sort of definition to how steep it is compared to the x-axis. When the slope is zero, this means that the point is either the top of a hill or bottom of a valley, the derivative is crossing the x-axis. The extrema are unfortunately not usually located on either of the axes; they are usually some point in one of the four quadrants. As a matter of definition, an extrema mathematically occurs when the slope of the polynomial at a given x location is 0. The power algorithm yields a derivative of $f'(x) =3x^2+7$, and when we solve for 0, then we find there are no real solutions for the derivative.

If a certain x-intercept has a multiplicity other than one, write out the multiplicity beside the point.

When a problem is given in any form other than standard, it is always best to turn it into that form.

This will leave you with three y values which you can pair up with their parent x value to obtain the extrema.

If you are graphing multiple equations on the same graph, it is best to color code the graphs or label them. These functions are excellent for demonstrating real life situations, such as banking trends. The polynomial $y = (x+6)(x-4)+5x^2$ is not factored because a 5x2 is being added rather than multiplied. Whenever you first begin observing a polynomial in preparation to graph, always examine the degree first.

Even degree functions have either an absolute minimum or maximum, so they don't go on forever in both directions, just one.

The slope is 3 everywhere since the value of y increases by three for every unit increase of x.

The job of the derivative is to tell us specifically the slope of every point on the curve4.

An imaginary root cannot be graphed as an x-intercept, and they do not provide much visual clue when the polynomial is graphed. The factored form allows us to set every factor to zero therefore making the entire thing possible to solve. To find the derivative, we use the power rule, $ax^n \rightarrow anx^{n-1}$ on each term when the polynomial is in standard form.

We therefore must establish a different viewing window, or in other words, different dimensions. The end behavior is totally dependent on the leading term of the polynomial function when simplified2. Therefore, there is a very large graphical connection between the function and its derivative.

Once you find the x part, simply substitute the values into the original polynomial, and the respective y parts will emerge. Nonetheless, they are still an important part of the function, and it is best to identify the imaginary roots whenever possible. It is a little more advanced than quadratic FOIL, but it is very much the same concept; every term in a factor gets multiplied by every term in another. An odd degree polynomial can have no extrema, but an even degree function must have at least one.

That doesn't mean there will be five x-intercepts; it just means there can be at most five x-intercepts, so expect that. Mark these down for later, and do not forget to indicate that the intercept (-3,0) has a multiplicity!

It is fully factored since there are no other factors that can be further simplified or factored.

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