12  Low-order polynomials

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Reading questions

Reading question 12.1 “Poly” means many, as in “polygon” (having many sides) or “polymer” (a molecule formed by many identical chemical units) or “polygram” (a “lie detector” which records many physiological signals at the same time). In contrast, “mono” means one as in “monocle” (a visual aid for one eye) or “monopoly” (which comes from the Greek for “one seller.”)

1. Provide a brief definition of “polynomial” in the mathematical sense. You definition should include the phrase “linear combination.”

2. Say how a “monomial” differs from a polynomial.

Reading question 12.2  

1. Which one of the following is not a monomial?

$x^{12}$       $t^0$       $x^{-2}$       $t^1$      

question id: not-a-mono-choice-1

2. Which one of the following is is a monomial?

$x^4$       $2+3x^2$       $\sqrt{x}$       $4^x$      

question id: not-a-mono-choice-2

3. Which one of the following is is a polynomial?

$3x^3-0.12^x$       $2.5+3.2x^2$       $6x-2\sqrt{x}$       $$-3x^{-1}+4+2.5x^1$$      

question id: not-a-mono-choice-3

Reading question 12.3  

1. All polynomials are linear combinations of functions. The “linear polynomial” refers to a specific set of functions. Which are they?

$x$ and $x^2$       $x^2$ and 1       1 and $x$       $x$ and $x^2$      

question id: linear-polynomial-1

2. Which of the panels in MOSAIC Calculus Fig 12.1 show linear polynomials?

A and E       D and E       B and E       B and A      

question id: linear-polynomial-2

3. Explain why $f(x)\equiv A(x-x_0)$ is a linear polynomial.

4. Explain why $g(t)\equiv A(x-x_0{)}^2$ is not a monomial.

Reading question 12.4 What makes a polynomial term an “interaction” term?

You can adjust the coefficient on it interactively.

It involves two monomials, each in a different input, being multiplied together.

It comes in the middle of a polynomial.

It has a coefficient of 1.

question id: interaction-term

Reading question 12.5 In the “low-order” polynomials described in the reading, what terms have the highest exponents?

linear terms

interaction terms

quadratic terms

cubic terms

question id: highest-order

Reading question 12.6 Which of the following is not a shape seen in a graph of a low-order polynomial in two variable?

Bowl       Barrel       Hilltop       Saddle      

question id: 2-var-shape

In the polynomial $$a_0+a_{x}x+a_{y}y+a_{xx}xx$$, what is the coefficient on the interaction term?

0       $a_{xy}$       $a_y$       $a_0$      

question id: coef-interaction

Reading question 12.7 Imagine a second-order polynomial in three inputs: $x$, $y$, and $z$, like this: $$b_0+b_{x}x+b_{y}y+b_{z}z+b_{xy}xy+b_{xz}xz+b_{xx}{x}^2+b_{yy}{y}^2+b_{z}z{z}^2.$$ All of the possible second-order (or less) terms are shown, except for one. Which term is missing?

the constant term

the quadratic term in $y$

the interaction between $z$ and $x$

the linear term in $z$

question id: missing-term

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