Non-Negative Polynomials

Task

A non-negative polynomial $f$ is a polynomial which never takes negative values, that is, $f(x) \geq 0$ for all real values of $x$.

1. Decide which of the following polynomials are non-negative:  $$ x^2\qquad\qquad x^2-1 \qquad\qquad x^3\qquad \qquad 100000-x^2\qquad \qquad mx+b$$
   In the last part, consider various possibilities for $m$ and $b$.
2. Show that if $g$ is a polynomial, then $g^2$ is a non-negative polynomial.  Use this fact to generate some non-negative polynomials.
3. Are all of the coefficients of a non-negative polynomial necessarily positive?
4. Is there a non-negative polynomial which has all negative coefficients?
5. Find a non-negative polynomial which is not the square of another polynomial.

IM Commentary

Polynomials constitute a rather subtle point in the common core framework -- whereas a vast majority of the time we think of $f(x)=x^2+1$ as a function, the standards surrounding polynomial arithmetic have students learn to treat these as objects in and of themselves, to be manipulated algebraically using much the same rules as we have for integer arithmetic. This difference in perspective is important, as statements like the Fundamental Theorem of Algebra can be viewed as statements about the algebraic structure of the set of polynomials, and not about their interpretations as functions. This task could be thought of as a transition from one viewpoint to the other -- students begin reasoning graphically with them, but are slowly led to more directly reasoning with them as polynomials, thinking about the process of adding and multiplying them, and reasoning with their degrees.  For example, students are implicitly asked to recognize that the square of a polynomial is another polynomial, a problem which doesn't admit a graphical solution.

The task helps foster student understanding of the analogy described in the standard -- "Understand that polynomials form a system analogous to the integers..." -- in addition to having the same arithmetic operations available, there are many other instances in which integers and polynomials share common properties.  In part (b) of this task, for example, we learn that much like the square of an integer (or real number) is non-negative, so is the square of any polynomial non-negative.

Further questions about non-negative polynomials abound, which teachers could use with more advanced students:  does a non-negative polynomial have to have even degree?  Can a non-negative polynomial have a negative leading coefficient?  Is the sum of two non-negative polynomials again a non-negative polynomial?  As a couple of related "bonus facts" for teachers to consider sharing with interested students, it is a remarkable fact that every positive polynomial is the sum of the squares of two polynomials. On the other hand, the statement is false for polynomials of two variables! The polynomial $x^4y^2+x^2y^4-3x^2y^2+1$ is non-negative, but is not the sum of squares of polynomials. For more information, see the Wikipedia entry for positive polynomials.

Solution

1. Since for any real value of $x$, the quantity $x^2$ is non-negative, the polynomial $x^2$ returns only non-negative values, and so is a non-negative polynomial.  On the other hand, $x^2-1$ is not non-negative as $0^2-1=-1<0$.  The polynomial $x^3$ returns negative values when $x$ is negative, so is not non-negative, and $1,000,000-x^2$, while non-negative on most standard domains on a graphing calculator, is negative for any value $x>1000$ (or $x<1000$).  Finally, a linear function $mx+b$ always has positive and negative values, unless $m=0$.  So the only non-negative linear function are those with $m=0$ and $b \geq 0$, i.e., the non-negative constant polynomials.

2. Suppose we set $h=g^2$. Then since a product of two polynomials is again a polynomial, $h$ is also a polynomial.  Also, since the square of any real number is non-negative, for any $x$ we have $h(x)=(g(x))^2\geq 0.$  This shows that $h$ is a non-negative polynomial.
   
   This makes it straightforward to find examples of non-negative polynomials: we take any polynomial we want, e.g., $g(x)=x^3+x$, and square it to get the non-negative polynomial $$x^6+2x^4+x^2.$$ This particular polynomial can also be seen to be non-negative because its only terms are even powers of $x$.

3. No.  While squaring a polynomial makes all of its values non-negative, its coefficients may or may not be negative.  For example, if $g(x)=x-1$, then its square, $x^2-2x+1$, is a non-negative polynomial (by the previous part) with a negative coefficient of $x$.  Many other examples exist as well.

4. No.  If all of the coefficients were negative, then plugging in any positive value for $x$ would result in a negative value of the polynomial.

5. Finding an example requires some playing around with non-negative polynomials and the process of squaring a polynomial.  One simple example is given by $h(x)=x^2+1$ (or $h(x)=x^2+c$ for any $c>0$), which is noteworthy for having no real roots.  This degree 2 polynomial could not be the square of a polynomial, since it would have to be the square of a degree 1 polynomial, and every degree one polynomial has a root (which would then also be a root of the quadratic).
   
   As a more algebraic alternative, we could consider the coefficients of a linear polynomial $g(x)=mx+b$ whose square was $x^2+1$.  Comparing coefficients,
   $$x^2+1=(mx+b)^2=m^2x^2+2mbx+b^2.$$
   For this to hold, we would need $m^2=1$, $b^2=1$, and $mb=0$, which is impossible since $mb = 0$ forces at least one of $m$ or $b$ to be 0.