While polynomials may seem very different on the surface, we can see through various explorations that they exert many similar characteristics.
For instance, we can see that polynomials of even degree have similar end behavior and follow a pattern which allows us to predict what higher powered polynomials will resemble. In the following graphic, we can see as the degree increases, the midsection “flattens”. This is the most obvious difference between the polynomials. We can also see the locus of points for all of these polynomials have common points of (1,1) and (-1, 1). You might ask why this is the case. If we perform the inverse operation of any even-degree polynomial in order to solve for a particular value, we can see the nth root of 1 is always 1 or -1. This explains the common points. However, we can also see, for positive coefficients, the end behavior is rising on both sides towards positive infinity.
How do we know the end behavior is increasing towards positive infinity? How do we know the graph will never come back towards the x-axis?
The best way to satisfy this curiosity is to test points. We should be able to pick any arbitrary point, magnitude irrelevant, and see the direct relationship (as one variable increases, so does the other). For instance, let’s consider x=1,692 and substitute it in for y=x^4. We yield y= and see when x is a large number, y is even larger. This makes sense since a number to a power will yield a result much larger than itself. Perhaps you ask, what about really small numbers, such as negative values? Think about what happens when you raise a negative number to an even power- the result is always positive. We can safely conclude that polynomials of even-degree will always increase towards positive infinity.
What happens when we have a negative leading coefficient?
We see a similar behavior to the positive leading coefficient except the graph has been reflected across the x-axis. We see the graphs still “flatten” as the degree increases and we have common points which are reflected along with the graph. We the locus of points are now located at points (1, -1) and (-1, -1). This happens for the same reason as above, but now there is a negative sign preceding.
What happens to the even degree polynomials if we make the coefficient a number other than 1?
Let’s consider a coefficient of 2. In the following example, I am only comparing y=x^2 for simplicity sake.
As you can see, the basic shape and end behavior of the polynomial has not changed. What has changed? We can see the major difference is how much the graph has “shrunk”. It seems that it is “skinnier” than the original. Furthermore, y=2x^2 is half the size of y=x^2, or it has been vertically stretched by a factor of 2. Because all even-powered polynomials behave in a similar fashion, we can conclude that we would see similar results with yx^4, y=x^6, etc.
What about coefficients higher than two? What do you think will happen to the graph?
Let’s try a coefficient of 10.
The pink line represents y=x^4, red is y=2x^4, and blue is y=10x^4. As you can see, even with other even-degreed polynomials, the larger the coefficient, the “skinnier” the graph becomes. This makes sense because we are now stretching the graph vertically by a factor of 10 this time instead of 2.
Let’s consider polynomials of odd-degree. Again, we can see that odd-degree polynomials exhibit similar characteristics to each other and we can predict what higher degree polynomials will look like.
Unlike the even degree polynomials, we see end behavior in opposing directions. The left side of the polynomial decreases towards negative infinity while the right side increases towards positive infinity. Much like the even degree polynomials, we see the locus of points for all odd-degree polynomials have common points of (1,1) but instead (-1,-1). The reason the common point changes to (-1, -1) is because the nth root of -1 is -1 for all odd roots.
How do we know the end behavior “behaves” like this all the time and not return to the x-axis?
Much like the even-degree behavior, we should test points. Much like the even degreed polynomial, if we select an arbitrary number and substitute the value into y=x^3, we will again see a direct relationship. As the value of x increases, the value of y will also increase. Furthermore, the smaller the value of x, the smaller the value of y will become. For instance, let’s consider x=-4. If we (-4)^3, we get an even smaller result of -64. While this is not a proof, we can safely assume any real number will also exhibit this quality.
What happens when we have a negative leading coefficient in odd-degree plynomials?
We see a similar trend as the even degree polynomial with a negative leading coefficient. It too has been reflected across the x-axis and the common locus of points have also been reflected. The common points are now (1, -1) and (-1, 1) since the negative preceding affects the final outcome. The end behavior on an odd degree polynomial with a negative leading coefficient has the left side increasing towards positive infinity while the right side is now decreasing towards negative infinity.
What happens to odd degree polynomials when the coefficient is a number other than 1?
Much like the even-degreed polynomials, we should similar results. The coefficient should only affect the stretch or compression of the graph, but not the end behavior. Let’s first investigate a coefficient of 3.
Again, we see y=3x^3 (red line) is seemingly “skinnier” and has been vertically stretched by a factor of 3. We should see a similar trend for larger coefficients. For instance, let the coefficient be 10 again.
We see that we were exactly right. Y=10x^3 (blue line) is even “skinnier”, or vertically stretched by a factor of 10.
We have now thoroughly explored various values of possible leading coefficients for even and odd degree polynomials. Perhaps we should consider how the coefficients of the following terms affect the behavior of the graph.
What do you think the middle terms control?
To find out, let’s keep the leading coefficient a constant one and the constant term zero.
Again, let’s first consider even-degreed polynomials.
The following graph represents y=x^2 (pink), y=x^2+x (red), and y=x^2+3x (blue). We can see that the middle term affects the location of the vertex, but does not stretch or compress the parabola.
How does the sign of the middle term affect the graph?
Let’s consider y=x^2 (pink), y=x^2-x (red), and y=x^2-3x (blue):
The middle term seems to have the opposite effect and translate the vertex left for positive values and right for negative values. Is this true for higher order polynomials? Let’s consider y=x^10, y=x^10-x, y=x^10-3x:
Again, this middle term affects the location of the vertex in a somewhat opposing manner.
Let’s briefly look at odd-degreed polynomials.
We can see a similar effect of the next middle term with these polynomials:
Y=x^3 (pink), y=x^3-x^2 (red), y=x^3-3x^2(blue)
y=x^3 (pink), y=x^3+x^2 (red), y=x^3+3x^2 (blue)
With odd-degreed polynomials, we can see the next middle term effects the graph by translating one of the “hills” and making the “hill” much steeper.
If we again look at a cubic polynomial, what happens if the quadratic term remains constant, but instead the linear term changes?
As you can see, the linear term has a similar effect on the cubic graph as the quadratic term. It too affects how large the “hills” and “valleys” of the cubic polynomial become. Essentially, the larger the linear coefficient becomes, the “straighter” the cubic function appears to be. This includes the assumption that negative coefficients are considered to be “small” numbers. The charcoal gray equation has the most significant valleys and also has the smallest linear coefficient value (y=x^3+x^2-5x). The “straightest” cubic equation appears to be the yellow equation and it has the largest linear coefficient (y=x^3+x^2+5x).
What conclusions can we make about the final constant term?
The final constant term has n roots which are all factors of the constant term. This is because (x-x1)(x-x2)(x-x3)…(x-xn) expanded will become an nth degree polynomial and constant x1x2x3…xn.
Suppose we have a simple cubic polynomial y=(x-1)(x-2)(x-3). Since this cubic is in factored form, we know the roots are x=1, 2, and 3. If we expand this cubic, we yield a result of y=x^3-6x^2+11x-6. As you can see, the constant term of the expanded form is the same as the product of factors.
We can consider a similar quartic polynomial problem. Suppose y= ; so the roots would be x=-4, 3, -2, and -5. The expanded form yields . Again, we can see the product of the factors matches the constant term. This makes sense because using the distributive property would have us multiplying the last terms of each factor.
What happens when we add polynomials together? Will the result also be a polynomial? Why or why not?
If we add two polynomials of the same degree, we will of course get another polynomial. This is because we are simply combining like terms and changing the shape of the polynomial (stretch or compression, possibly a reflection). When the polynomials are of the same degree, we are not drastically changing the exponents or x so that it would no longer be a polynomial (for instance, adding polynomials will not involve taking the square root of x). If we add polynomials of different degrees, we will still result in a polynomial, but it likely to take the shape of the higher power. For instance:
f(x)= 3x^2-x+2 and g(x)=3x^5+4x^2-1
f(x)+g(x) will yield a quintic polynomial. See below:
The blue function bears resemblance to the higher-degreed polynomial (in this case, g(x)).
Is the product of polynomials a polynomial? Why or why not?
The product of polynomials should yield a result of a higher-powered polynomial. Using the laws of exponents, when multiplying polynomials we are simply adding the exponents (using the product rule). Regardless of the polynomial, we should always maintain whole numbers as the exponents. This can quickly become a tedious manipulation due to complicated distributing, so let’s consider a simple example:
f(x)=x^4 and g(x)=3x^7
f x g (x)=3x^4+7=3x^11
The exponent is still a positive whole number, and therefore still meets the qualifications of being a polynomial.
Does the quotient of polynomials yield always yield a polynomial?
No, we can see using one such example, we can see through long division that the quotient involves a remainder. The remainder is not a polynomial. However, the remainder is an example of a rational function. A rational function is defined as the quotient of two polynomial functions.
What can we use polynomials for? Why are they important?
Many phenomena in the universe are best-fit using polynomials. Given the simplicity of polynomials and the nice curves they produce, they are ideal for approximating other more complex functions. If we are able to approximate these functions, then we can in turn make predictions about these particular phenomena and apply it.
What is the significance of the degree of a polynomial and how can we determine the number of roots a polynomial has based on the degree?
Through the Fundamental Theorem of Algebra, we can determine the number of roots a polynomial has. This theorem tells us that polynomial of nth degree has no more than n roots (real or complex) and must have at least one complex root. Generally speaking, a linear polynomial would have one root, a quadratic would have two roots, a cubic would have three roots, and so on.
Polynomials are particularly “nice” because they lack holes and asymptotes (“discontinuities”). Every point along the graph is differentiable (we are able to find the tangent line at every point). Furthermore, we are able to integrate just as “nicely” and approximate the area under the curves. This is precisely why polynomials are the choice of many scientists to predict behavior of various phenomena.