# Lesson 3

Introducing Polynomials

### Problem 1

Select all points where relative minimum values occur on this graph of a polynomial function.

A:

Point $$A$$

B:

Point $$B$$

C:

Point $$C$$

D:

Point $$D$$

E:

Point $$E$$

F:

Point $$F$$

G:

Point $$G$$

H:

Point $$H$$

### Problem 2

Add one term to the polynomial expression $$14x^{19}-9x^{15}+11x^4+5x^2+3$$ to make it into a 22nd degree polynomial.

### Problem 3

Identify the degree, leading coefficient, and constant value of each of the following polynomials:

1. $$f(x)=x^3 - 8 x^2 - x + 8$$
2. $$h(x)=2 x^4 + x^3 - 3 x^2 - x + 1$$
3. $$g(x)=13.2 x^3+3 x^4 - x - 4.4$$

### Problem 4

We want to make an open-top box by cutting out corners of a square piece of cardboard and folding up the sides. The cardboard is a 9 inch by 9 inch square. The volume $$V(x)$$ in cubic inches of the open-top box is a function of the side length $$x$$ in inches of the square cutouts.

1. Write an expression for $$V(x)$$.
2. What is the volume of the box when $$x=1$$?
3. What is a reasonable domain for $$V$$ in this context?

### Solution

(From Unit 2, Lesson 1.)

### Problem 5

Consider the polynomial function $$p$$ given by $$p(x)=7x^3 - 2x^2 + 3x+10$$. Evaluate the function at $$x=\text-3$$.

### Solution

An open-top box is formed by cutting squares out of an 11 inch by 17 inch piece of paper and then folding up the sides. The volume $$V(x)$$ in cubic inches of this type of open-top box is a function of the side length $$x$$ in inches of the square cutouts and can be given by $$V(x)=(17-2x)(11-2x)(x)$$. Rewrite this equation by expanding the polynomial.