Which of the following is the graph of a polynomial of even degree with a positive leading coefficient?
So the first thing we think about is the turning points. That’s because if we have an even maximum number of turning points, then this means the polynomial is gonna be of an odd degree. And that’s because the maximum number of turning points is always one less than the degree of the polynomial.
So, for example, if we have a quadratic, so that has a degree of two cause we’ve got 𝑥 squared, then this will have one turning point. And you see the shape of a quadratic is a curve with a maximum or minimum.
So therefore, if we’re looking for a polynomial with an even degree, then we’ll need an odd number of turning points. So therefore, we can rule out B. And that’s because as we can see, there’re four turning points. So we’ve got an even number of turning points. And we can also rule out C. And that’s because we’ve got two turning points. So therefore, again this is going to have an odd degree because it’s in fact cubic.
So now, how we’re gonna determine between A and D? Well, we can use a bit of information that tells us that we’re looking for a polynomial with a positive leading coefficient. So the leading coefficient will affect the shape of the graph of polynomial. If it’s positive and we’d have an even degree, then it’ll take a shape something like this. But the key aspect is that we are coming from high. So we’ve got a negative slope to begin with. And then it carries on in a form like this.
If it’s a negative leading coefficient and we have an even degree, then it’ll take a form something like this. This time, we have a positive slope to begin with. And then, it’ll carry on and take a shape a bit like this. So therefore, we can rule out graph D. That’s because it has a negative leading coefficient. So it takes one like a M shape that I showed.
So that means that the graph of the polynomial of an even degree with a positive leading coefficient is going to be graph A. And that’s because we have an odd number of turning points. So it’s gonna be an even degree. And it’s in the correct shape because it is in a W shape for this particular polynomial to show that it has a positive leading coefficient.