# Lesson 3

Introducing Polynomials

### Problem 1

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

Point \(A\)

Point \(B\)

Point \(C\)

Point \(D\)

Point \(E\)

Point \(F\)

Point \(G\)

Point \(H\)

### Solution

For access, consult one of our IM Certified Partners.

### 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.

### Solution

For access, consult one of our IM Certified Partners.

### Problem 3

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

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

### Solution

For access, consult one of our IM Certified Partners.

### 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.

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

### Solution

For access, consult one of our IM Certified Partners.

(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

For access, consult one of our IM Certified Partners.

(From Unit 2, Lesson 2.)### Problem 6

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.

### Solution

For access, consult one of our IM Certified Partners.

(From Unit 2, Lesson 2.)