Week 1: Polynomials – Answers

Week 1: Polynomials – Day 5

1. What is the domain of a polynomial function?

The domain of any polynomial function is all numbers, in interval notation (-∞ , ∞ ).

2. What is the range of a quadratic (degree 2) polynomial function?

Quadratic polynomials either have an absolute maximum or minimum; their graphs are parabolas that either open up (and have a minimum) or open downward (and have a maximum). So the ranges, the set of y-values, either look like (- ∞ , maximum] or [minimum, ∞)

3. What is the range of a cubic (degree 3) polynomial function?

All numbers. In interval notation (-∞ , ∞ ).

4. True or false: the product of two polynomials is a polynomial.

True. The product of two polynomials is a sum of terms of the form (coefficient)*(positive whole power of x), which is a polynomial.

An interesting question is what about the roots of the product. The answer is that the roots of individual polynomials are also the roots of the product. Think about why that is true.

5. True or false: the sum of two polynomials is a polynomial.

True. The sum of two polynomials is a sum of terms of the form (coefficient)*(positive whole power of x), which is a polynomial.

An interesting question is what about the roots of the sum. The answer is that all bets are off; knowing the roots of the first two polynomials doesn’t tell you about the roots of the sum.

Think about adding x and 1, for example. The polynomial f(x) = x has one root at zero. The polynomial g(x) = 1 has no roots. The polynomial f + g = x + 1 has one root, at x = -1.


 

Week 1: Polynomials – Day 4

Cubic polynomials. All the graphs are sketched below the questions.

1. Sketch the graph of f(x) = x3.

2. Sketch the graph of g(x) = x(x2 + 1).

Bonus questions: what is the difference between f(x) and g(x), in the neighborhood of x = 0? How many real roots do f(x) and g(x) have?

Both f and g have one real root, at x = 0. In the case of f(x), the root has multiplicity 3; for g(x), the root has multiplicity 1. When you graph the two of them, there is different behavior near x=0. This is because the cube of a tiny number is super-tiny.

In other words, if x is near 0, such as 0.01 (one hundredth), f(0.01) is 0.000001 (one millionth) – super close to zero.

But g(0.01) = 0.01(1.0001) = 0.010001, which is farther from zero.

This means that f(x) has different behavior near zero – it runs to zero, and g(x) walks. Think about it.

3. Sketch the graph of h(x) = x(x – 3)2. Hint: how many real roots does h(x) have? What are the values of h(x) around and between the roots?

h(x) has two real roots: 0 and 3. h(-1) = -16. h(1) = 4. h(4) = 4.

4. Sketch the graph of p(x) = x(x – 1)(x – 2). Hint: how many real roots does p(x) have? What are the values of p(x) around and between the roots?

p(x) has three real roots: 0, 1, and 2. p(-1) = -6. p(1/2) = 3/8. p(3/2) = -3/8. p(3) = 6.

Note: sketching graphs of polynomials (without the benefit of calculus) means making judicious choices of points to compute and plot. I chose the roots and points to the left and right of roots. The idea is to use the fewest points to capture all of the features.

5. Sketch the graph of q(x) = -x(x – 1)(x – 2).

cubicgraphs


 

Week 1: Polynomials – Day 3

1. Make a quick sketch of the function:       f(x) = x2 – 4.

Desmos_Graphing_Calculator

2. Make a quick sketch of the function:       g(x) = (4 – x)2.

Desmos_Graphing_Calculator

3. Let h(x) = f(x)*g(x) = (x2 – 4)( 4 – x)2 . What kind of function is h(x)?

h(x) is a 4th degree polynomial. The leading term is x4.

4. How many roots does h(x) have?
h(x) has three distinct roots: 2, -2, and 4. (4 is a root with multiplicity 2).

5. Make a quick sketch of the graph of h(x). Hint: you can find all the real roots of h(x). Test the values of points between and around the roots; e.g. h(-3), h(0), h(3), h(5).

h(-3) = 5*49 = 245; h(0) = -4*16 = -64; h(3) = 5*1 = 5; h(5) = 21*1 = 21
h(-2) = h(2) = h(4) = 0

sketch_h


 

Week 1: Polynomials – Day 2

answers1-3

 

answers4-5.docx


 

Week 1: Polynomials – Day 1

1. If P(x) is a polynomial of degree 7, and Q(x) is a polynomial of degree 2, what kind of function is f(x) = P(x)*Q(x)?

f(x) is a polynomial function of degree 9.

2. Let P(x) be a polynomial of degree 7. What is the highest number of real roots (zeroes) that P(x) could have?

P(x) has at most 7 real roots.

3. Let P(x) be a polynomial function of degree 7. What is the lowest number of real roots (zeroes) that P(x) could have?

A polynomial of odd degree has at least one real root. P(x) has degree 7, and therefore at least one real root.

4. Let Q(x) be a polynomial function of degree 4. What is the lowest number of real roots (zeroes) that Q(x) could have?

A polynomial of even degree can have zero roots. The minimum number of roots Q(x) could have is zero.

5. Let f(x) = P(x) = x*x*x*x*(x – 1)*(x – 2)*(x – 3). How many distinct real roots does f(x) have?

4 distinct roots: 0, 1, 2, and 3.