Understanding equivalent forms of polynomial functions to answer questions in context
About this video
The video compares three ways to write a polynomial: standard form, vertex form and factored form. Standard form makes the y-intercept (initial value) easy to read, vertex form shows the vertex, and factored form shows the zeros where the graph crosses the x-axis.
You'll be able to
- recognise standard form, vertex form and factored form of a polynomial.
- read the y-intercept from standard form, the vertex from vertex form and the zeros from factored form.
- choose the most useful form for a question in context.
Key words
- Standard form
- A polynomial written as a sum of terms in decreasing powers, like 2x^2 - 3x + 9.
- Vertex form
- A form like y = a(x - h)^2 + k that shows the vertex directly.
- Factored form
- A polynomial written as a product, like a(x - r1)(x - r2), showing its roots.
- Y-intercept
- The value of y when x = 0; often the starting value in context.
Transcript
- [Instructor] So let's talk a little bit about the different ways that we can represent a polynomial. So first is standard form. Standard form. And this is a situation we could, if I were to say an arbitrary and nth degree polynomial, you'd have your first coefficient a sub n times x to the nth power plus the next coefficient, a sub n minus one times x to the n minus one power, all the way to a sub one times x plus a sub zero.
Now, to make this a little bit tangible, I could do this when n is two, if we were dealing with a quadratic. And an example of that would be, and actually, let me write it this way. So we could say y is equal to all of that. And in the particular form, as a quadratic, I could write y equal, and I'm just putting random numbers in here.
This is nothing special about this quadratic. 2x to the second power, that's what makes it a quadratic,
- 3x plus nine. So if you think about it, you know, these last three terms here is essentially what I have represented here. Now, another form that you could see, and then I'll talk a little bit about when one might be better than the other, is vertex form. Vertex form.
And if we're speaking generally about any polynomial, vertex form would be y is equal to some constant times x minus some other constant to the degree of the polynomial plus some constant. If I were to write a quadratic in this form, and I'll do a different quadratic because I'm not going to take the time to complete the square of this first one in this video, it could be something like y is equal to minus 3 times x plus four squared minus four.
To be clear, if I wanted to, I could rewrite this quadratic in standard form. These are just different ways of algebraically representing the same thing. And last but not least, you also have factored form. Factored form. And the general factored form would be y is equal to a times x minus one of the roots times x minus another one of the roots.
You could imagine when x is equal to R1, this would make this term or what's in the parentheses equal to zero. And that would make, when you multiply anything times zero, it would make everything equal to zero. So r1, r2, and so on and so forth. Those are, when x takes on those values, any of those values, our function is going to be equal to zero.
And just like I did before, if I were to say a quadratic in that form, it'd be something like, let's say four times x plus one times 2x minus three. Now, again, these are all different functions, but I could rewrite this blue function in vertex form or I could rewrite it in standard form by algebraically manipulating it.
Now, the key question is, when is one of these forms better than the others? Well, generally speaking, standard form is good if you want to figure out the initial conditions. So for example, if you wanted to say, "All right, when x equals zero, what's y?" Or if we were to say if this was f of x is equal to this, when x is equal to zero, when we're just starting off, what is y?
And because these terms would just go away and you would clearly see that our y-intercept is nine. And that could be useful in business problems or physics problems. Let's imagine that you have a quadratic that looks something like this. So in a physics, if you're throwing something, that might be your initial height that you're throwing it from.
In a business situation, that might be your initial profit or your initial cost, depending on how you're doing it. And oftentimes people find it easiest if you're just trying to figure out, what is the value of the function for a given x, this standard form is oftentimes what people like.
Now, you could also do that in these other two, but that's where people gravitate. Now, what about vertex form? Well, as the name implies, it's a very useful way of figuring out the vertex. In this situation, the vertex is here. What is the x value of that vertex?
Well, the x value is what's going to make this piece right over here equal to zero. In this case, x would be equal to -4. And then last but not least, what's useful about factoring, or factored form? Well, you can figure out where the polynomial, in this case the quadratic, intersects the x-axis.
So if you wanted to figure out the zeros, if you wanted to figure out this point, in a physics context or in a business context, this point over here might not be of interest, but if you wanted to figure out these points, then factored form is very, very, very useful.
So that's just the big takeaway. These all represent different ways that you can represent a polynomial. And in general, the last one, factored form, useful for roots where you're intersecting the x-axis, vertex form, useful for the vertex, and standard form, useful for the y-intercept or just evaluating values.
Although I think evaluating values vertex form is also reasonably straightforward.
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