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Quiz about The Mis Adventures of Miss Polly Nomial

The (Mis) Adventures of Miss Polly Nomial Quiz

This is a quiz all about polynomials and their properties. Enjoy!

A multiple-choice quiz by Mrs_Seizmagraff. Estimated time: 7 mins.

Author
Mrs_Seizmagraff

Time
7 mins

Type
Multiple Choice

Quiz #
183,640

Updated
Dec 03 21

# Qns
15

Difficulty
Tough

Avg Score
9 / 15

Plays
3620

Awards
Editor's Choice

One at a Time - Single Page - Timed Game

Question 1 of 15

1. Miss Polly Nomial decided to build a water slide in the shape of the graph of her favourite polynomial function. She asked her chief architect if this was a good idea, what did he say? Hint

No - the water slide rider could fall down one of the cracks in the graph

Yes - all polynomial functions are continuous

Yes - the asymptotes in the graph would make for a fun water ride

No - polynomial functions are flat, so the rider would have nowhere to go

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Question 2 of 15

2. One of Polly's favourite things to do is to go down the slide in the park; she likes it because it has constant (non-zero) slope. What sort of polynomial function is the slide?

Hint

Constant function

Cubic function

Quadratic function

Linear function

3. Polly's cousin, Connie Conic, came over for a visit. She brought with her one of her favourite conic sections. As it turned out, Polly had the same shape in her polynomial functions collection! What conic would this be? Hint

Parabola

Circle

Hyperbola

Ellipse

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Question 4 of 15

4. Polly's favourite song is "Holding out for a Zero" by Tonnie Byler. In Polynomial Land, zeros are VERY important. One of Polly's polynomial functions has degree 5. What is the MAXIMUM number of zeros her function can have? Hint

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Question 5 of 15

5. Polly's little sister, Mono Nomial, has a degree 4 function as a pet. What is the MINIMUM number of real zeros her function can have? Hint

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Question 6 of 15

6. If you listen to the lyrics of "Holding out for a Zero", Tonnie will tell you how to determine if a given number (say "a") is a zero of a given polynomial function (say "p(x)"). What will Tonnie tell you? Hint

If p(a) = 0, then "a" is a zero of p(x)

If p(-a) = 0, then "a" is a zero if p(x)

If p(0) = a, then "a" is a zero of p(x)

If p(0) = -a, then "a" is a zero of p(x)

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Question 7 of 15

7. Sometimes, Polly only cares about finding the rational roots to one of her functions. She has one such function now, all with integral coefficients, of the form y = ax^n+bx^(n-1)+...+c, where c is not zero. Which of the following could NOT be a possible rational root of this function? Hint

Any factor of "c" (the constant term)

-1

A fraction of the form 1/m, where "m" is a divisor of "a" (the leading coefficient)

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Question 8 of 15

8. On rainy Sunday afternoons, Polly spends her time factoring polynomials. She is working on one such polynomial now, p(x). Using the Rational Root Theorem, she has determined that p(-2/3) = 0. According to the Factor Theorem, which of the following is a factor of p(x)? Hint

2x - 3

3x - 2

3x + 2

2x + 3

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Question 9 of 15

9. Polly was #1 in her calculus class. She knows that if a polynomial function has a constant second derivative, and that second derivative is positive, then the graph of the function is always: Hint

decreasing

concave down

concave up

increasing

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Question 10 of 15

10. Polly has a polynomial function greater than degree 2, p(x), and she knows that there is a number "c" such that p'(c)=0 (the first derivative of the function evaluated at c equals zero) BUT p(c) is NOT zero. What does Polly know about the graph of p(x) for sure? Hint

At x=c, the graph has a minimum or maximum

The function has no real roots

At x=c, the graph has an inflection point

The function has complex roots

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Question 11 of 15

11. Polly is skipping rope with two of her best friends, Max and Min. She notices that the shape of the skipping rope is like a polynomial function, with Max (at one end) at the point "a" and Min (holding the other end of the rope) at the point "b". Max and Min are holding the rope at the same height, so if the rope was the function p(x) that means p(a)=p(b). "I'm on a Rolle!" screamed Polly. What can we say about p(x) in the interval (a,b)? Hint

There is some "c" in (a,b) with p'(c)=0

The graph of p(x) is increasing on (a,b)

There is some "c" in (a,b) with p(c)=p(a)=p(b) (c not equal to a or b)

The graph of p(x) is concave down on (a,b)

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Question 12 of 15

12. Polly knows there are many different theorems that tell us many beautiful things about polynomial functions. One such theorem says, on an interval (a,b), there will ALWAYS be some number "c" such that p'(c) = p(b)-p(a)/b-a. Polly likes the result of this theorem, despite the fact that this theorem can be "nasty". Which theorem is this? Hint

Intermediate Value Theorem

Rolle's Theorem

Mean Value Theorem

Fundamental Theorem of Calculus

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Question 13 of 15

13. Polly likes solving polynomial equations like p(x)=0. For quadratic equations, she solves them by completing the square. For cubic and quartic (degree 3 and degree 4, respectively) equations, she tries other methods - if those fail, she can always rely on certain formulas to find the roots for her. Can she use a formula to find the roots of a quintic (degree 5) polynomial? Hint

Yes - the formula is very complex though, and a computer program is needed to determine the roots

Yes - the formula was discovered in the 1800s and can be done by hand

No - it is impossible to construct such a formula

No - although the formula exists, Polly would need a PhD to understand it

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Question 14 of 15

14. Polly has bad dreams sometimes after she eats too much Mexican food right before bedtime. One of her scariest goes like this: she is taking her favourite polynomial function (degree n) out for a walk, and she runs into the big bad differential operator, who threatens to differentiate her pet function until it is 0. How many times would the operator have to differentiate Polly's function to make the function 0? Hint

n+1

n-1

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Question 15 of 15

15. Polly loves her life; the world of polynomial functions is a happy place where all children can play together, regardless of what colour or religion they are. This was decreed by King Riemann. Unfortunately, not all function families are like this. To what function operation am I referring? Hint

Inversion

Differentiation

Integration

Composition

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Oct 23 2024 : BigJim67: 5/15
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Quiz Answer Key and Fun Facts

1. Miss Polly Nomial decided to build a water slide in the shape of the graph of her favourite polynomial function. She asked her chief architect if this was a good idea, what did he say?

Answer: Yes - all polynomial functions are continuous

All polynomial functions are indeed continuous, and would have no cracks or asymptotes, so there would be no danger for the rider. Although constant functions (p(x) = c) are indeed both flat and polynomial, a cubic is also polynomial and there would certainly be no danger in not having anywhere to go on a cubic!

2. One of Polly's favourite things to do is to go down the slide in the park; she likes it because it has constant (non-zero) slope. What sort of polynomial function is the slide?

Answer: Linear function

I hope the term "constant" didn't trick you into choosing "constant function". Only linear functions have constant slope. (Technically, constant functions also have constant slope - zero slope. However I stipulated that Polly's slide had non-zero slope. Wouldn't a zero-slope slide be boring?)

Answer: Parabola

The only conic section that is also a polynomial function (or can be) is the parabola, y=x^2. The hyperbola can also be a function (such as when the equation is y = 1/x). The circle and the ellipse cannot be functions.

Answer: 5

By the Fundamental Theorem of Algebra, a polynomial of degree n can have at most n distinct zeros.

5. Polly's little sister, Mono Nomial, has a degree 4 function as a pet. What is the MINIMUM number of real zeros her function can have?

Answer: 0

Even functions (degree 2, 4, 6, 8, ...) do not have to have any real zeros. For example, the function y=x^4 + 7 does not have any real zeros. This means that the graph of the function never crosses the x- axis. Odd functions, however (1, 3, 5, ...) MUST have at least one real root.

Answer: If p(a) = 0, then "a" is a zero of p(x)

This only works for real zeros, and is a consequence of the Remainder Theorem. The Remainder Theorem says that when a polynomial is divided by (x-a), the remainder is p(a). If p(a)=0, this means (x-a) divides p(x) evenly (Factor Theorem) and a is zero.

Answer: 0

The Rational Root Theorem for integral polynomials says that any possible rational root is of the form m/n, where "m" divides the constant term and "n" divides the leading term. Since 1 and -1 divide both, that means -1 and 1/m COULD POSSIBLY be roots. There is no way 0 could be a root because p(0) = c and c is not 0.

Answer: 3x + 2

The factor theorem says that if P(a) = 0, then (x-a) is a factor of p(x). Knowing that p(-2/3)=0, this means x-(-2/3) is a factor, or equivalently (x + 2/3) is a factor; or equivalently (3x+2) is a factor.

9. Polly was #1 in her calculus class. She knows that if a polynomial function has a constant second derivative, and that second derivative is positive, then the graph of the function is always:

Answer: concave up

Think of the graph of the quadratic y=x^2. Positive second derivative (by the second derivative test) means the graph is always concave up. Note that the graph can be both increasing and decreasing and still concave up.

Answer: The function has complex roots

The situation described above corresponds to a "hiccup" in the graph, and indicates the presence of complex roots. There COULD be a minimum or a maximum at the point x=c, but we do not know that for sure. Consider y=x^3+1. At the point x=0, the derivative is 0, but there is neither a minimum nor a maximum there.

The same is true about inflection points. All we know for certain about this polynomial is that it has complex roots.

Answer: There is some "c" in (a,b) with p'(c)=0

The big hint was "Rolle" - as in "Rolle's Theorem". Rolle's Theorem says that if a function is continuous and differentiable on an interval (a,b) (which we know all polynomial functions are), and p(a)=p(b), then there is some "c" in that interval with p'(c)=0. Graphically, that means that the function hits a critical point (either a min, max, or inflection point) in that interval (or is constant).

Answer: Mean Value Theorem

What a broad hint! This theorem is "nasty" - ie, it is "mean"! The Mean Value Theorem is actually regarded as one of the most important theorems in mathematics, and can be used to prove many other wonderful results, such as Taylor's Theorem. Rolle's Theorem and the Intermediate Value Theorem are special cases of the Mean Value Theorem.

Answer: No - it is impossible to construct such a formula

Although formulas do exist for the general solution of polynomials of degree 2, 3, and 4, Abel proved in the 1800s that no such formula can ever exist for the general solution of equations of degree 5 or higher. This makes Polly quite depressed and she dislikes Abel very much.

Answer: n+1

The operation of differentiation reduces the degree of a polynomial function by 1. Starting with a function of degree n, after n derivatives the function will be constant (ie, degree 0). After n+1 derivatives, the function would be 0 (since the derivative of any constant number is 0).

Answer: Integration

While the other three certainly apply to polynomial functions, the correct answer here is integration. Under Riemann's definition, all polynomial functions are integrable. Some classes of functions are not as easily integrated; this lead to the development of measure theory and Lebesgue integration in the 1900s.

Source: Author Mrs_Seizmagraff

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