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NonNegative Polynomials
Task
A nonnegative polynomial $f$ is a polynomial which never takes negative values, that is, $f(x) \geq 0$ for all real values of $x$.
 Decide which of the following polynomials are nonnegative: $$ x^2\qquad\qquad x^21 \qquad\qquad x^3\qquad \qquad 100000x^2\qquad \qquad mx+b$$
In the last part, consider various possibilities for $m$ and $b$.  Show that if $g$ is a polynomial, then $g^2$ is a nonnegative polynomial. Use this fact to generate some nonnegative polynomials.
 Are all of the coefficients of a nonnegative polynomial necessarily positive?
 Is there a nonnegative polynomial which has all negative coefficients?
 Find a nonnegative 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 nonnegative, so is the square of any polynomial nonnegative.
Further questions about nonnegative polynomials abound, which teachers could use with more advanced students: does a nonnegative polynomial have to have even degree? Can a nonnegative polynomial have a negative leading coefficient? Is the sum of two nonnegative polynomials again a nonnegative 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 $x^4y^2+x^2y^43x^2y^2+1$ is nonnegative, but is not the sum of squares of polynomials. For more information, see the Wikipedia entry for positive polynomials.
Solution

Since for any real value of $x$, the quantity $x^2$ is nonnegative, the polynomial $x^2$ returns only nonnegative values, and so is a nonnegative polynomial. On the other hand, $x^21$ is not nonnegative as $0^21=1<0$. The polynomial $x^3$ returns negative values when $x$ is negative, so is not nonnegative, and $1,000,000x^2$, while nonnegative 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 nonnegative linear function are those with $m=0$ and $b \geq 0$, i.e., the nonnegative constant polynomials.

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 nonnegative, for any $x$ we have $h(x)=(g(x))^2\geq 0.$ This shows that $h$ is a nonnegative polynomial.
This makes it straightforward to find examples of nonnegative polynomials: we take any polynomial we want, e.g., $g(x)=x^3+x$, and square it to get the nonnegative polynomial $$x^6+2x^4+x^2.$$ This particular polynomial can also be seen to be nonnegative because its only terms are even powers of $x$. 
No. While squaring a polynomial makes all of its values nonnegative, its coefficients may or may not be negative. For example, if $g(x)=x1$, then its square, $x^22x+1$, is a nonnegative polynomial (by the previous part) with a negative coefficient of $x$. Many other examples exist as well.

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.

Finding an example requires some playing around with nonnegative 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.
NonNegative Polynomials
A nonnegative polynomial $f$ is a polynomial which never takes negative values, that is, $f(x) \geq 0$ for all real values of $x$.
 Decide which of the following polynomials are nonnegative: $$ x^2\qquad\qquad x^21 \qquad\qquad x^3\qquad \qquad 100000x^2\qquad \qquad mx+b$$
In the last part, consider various possibilities for $m$ and $b$.  Show that if $g$ is a polynomial, then $g^2$ is a nonnegative polynomial. Use this fact to generate some nonnegative polynomials.
 Are all of the coefficients of a nonnegative polynomial necessarily positive?
 Is there a nonnegative polynomial which has all negative coefficients?
 Find a nonnegative polynomial which is not the square of another polynomial.
Comments
Log in to commentelliot says:
7 monthsthis is a great task  is it really aligned to the standard: are you asking learners to add, subtract, or multiply polynomials? are they given the opportunity to explore the closure of polynomials under the rlementary operations? (no, no, maybe this belong somewhere else)
elliot says:
7 months*elementary