Hi Mike. How would you do practise test 8, section 3 question 11? Can you explain the whole polynomials to an odd/even degree etc. The collegeboard answer was a little confusing

Thank you!

Sure. The first thing you need to notice is that the graph has x-intercepts at –3, 0, and 2, which means you need factors of x + 3, x, and x – 2. That lets you eliminate choices C and D. I bet you got that part without much trouble.

After that, you need to choose between choice A, y=x(x-2)(x+3) and choice B, y=x^2(x-2)(x+3). The only difference between the choices, of course, is that one has just x while the other has x^2. Expanded, choice A will begin with an x^3 term, while choice B will begin with an x^4 term.

Basically, you need to know that the highest power of a polynomial tells you whether its y-values will extend to positive and negative infinity, or just to one or the other. If the highest power is odd, the polynomial will go to infinity in both directions, like so:


If the power is even, it’ll only go to infinity in one direction, like so:


Because the graph in the provided figure only goes towards positive infinity on the y-axis, you know you need the choice that gives you an even powered polynomial. That’s choice B!


Comments (1)

The College Board explains this differently:

“At x=0, the graph touches the x-axis but doesn’t cross it. This means that x, as a factor of f, is raised to an even power. If x were raised to an odd power, then the graph would cross the x-axis.”

This seems to suggest a rule about odd and even powers, but, your graphs seem to contradict this. Can you explain further?

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