Sunday, March 7, 2021

Algebra 2 PARCC: Factor an unknown polynomial


The following multiple-choice question, explained here in hopes of helping algebra 2 students and their parents in Maryland prepare for the PARCC test near the end of this school year, appears on the released version of the PARCC Algebra 2 sample test released following the 2016 test (#10).

chalkboard with polynomial formulas sketched

For an unknown polynomial function p(x), p(4) + 2 = 2. Which binomial is a factor of p(x)?

A.   x – 2
B.   x + 2
C.   x – 4
D.   x + 4

Correct answer and references

Correct answer: (C) x – 4

The high school math standards in the Common Core, “Functions” section (HSF.IF.C.7.C), require students to be able to “graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior.”

The question is aligned nicely to this math standard.

Because the zero is not given directly but must be discerned from an algebraic operation on the function, the question would fall into a depth of knowledge 2 range, even though the algebra is elementary.

Sample solution strategy (there are others)

Try the easy way first.

If evaluating the polynomial, whatever it is, at x = 4 and then adding 2 gives 2, that must mean that the value of the polynomial at that point is 0. Lucky for you.

If the value of p(4) is 0, that means 4 must be a root of the polynomial, or a zero. And that means that (x – 4) is a factor of the polynomial. It has to be.

In other words, this polynomial could do all kinds of crazy things, but at x = 4, it has to hit the x axis, giving a y value of 0. Maybe the polynomial looks like this:

a graph of a polynomial with root at x = 4

Maybe it looks like this:

a graph of a polynomial with root at 4

But whatever it looks like, wherever it wanders, if it crosses the x axis at 4, one thing you know for sure is that (x – 4) is one of the factors.

If you think about this question for a little while, pondering the beauty and perfection of algebra and mathematics in general, you will quickly discover that the only way you can answer a question like this is if you know a zero of the polynomial.

That means, whenever this topic is tested in algebra 2, the question writer has to give you some way of figuring out what the zero or zeroes are for the polynomial. That’s the only way you can determine what one of the factors is for an unknown polynomial.

Keep in mind that polynomials may be represented graphically as well as algebraically. But however the unknown polynomial is described, just remember you need to know the zeroes.

Paul Katula
Paul Katula is the executive editor of the Voxitatis Research Foundation, which publishes this blog. For more information, see the About page.

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