Monthly Archives: October 2014

Week 1: Polynomials – Day 5

1. What is the domain of a polynomial function?

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

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

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

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

Week 1: Polynomials – Day 4

Cubic polynomials.

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?

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?

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?

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

Week 1: Polynomials – Day 3

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

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

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

4. How many roots does h(x) have?

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).

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)?

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

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?

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?

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?