Question

Write each polynomial function in standard form, then classify it by degree and number of terms and describe its end behavior.

y = 2x(x^2 – 3)(x^2 + 2)

Answer 1

• y = 2x^5 -2x^3 -12x
• degree 5, 3 terms, end behavior: (-∞, -∞), (+∞, +∞)

Explanation:

I like to form the product of something like this by multiplying binomial pairs first. The distributive property applies for that (as does FOIL).

... (x^2 -3)(x^2 +2) = x^2·x^2 + x^2·2 + (-3)·x^2 + (-3)·2

... = x^4 -x^2 -6 . . . . . the two x^2 terms combine

Now, it is a simple matter to multiply each term by 2x:

... y = 2x^5 -2x^3 -12x

_____

The highest-degree term has degree 5. There are 3 terms. (All are odd-degree, so this is an odd function, symmetrical about the origin.)

As with any odd-degree function (with positive leading coefficient) the overall shape has a positive slope (/), tending toward -∞ for large negative values of x, and tending toward +∞ for large positive values of x.