# Lesson 4

Combining Polynomials

### Problem 1

Here are two expressions whose product is a new expression, $$A$$.

$$\displaystyle (5x^4 + \boxed{\phantom{33}}x^3)(4x^{\boxed{\phantom{3}}} - 6) = A$$

Andre says that any real number can go in either of the boxes and $$A$$ will be a polynomial. Is he correct? Explain your reasoning.

### Problem 2

Lin divides the polynomial $$2x^2 - 4x + 1$$ by 4 and gets $$0.5x^2 - x + 0.25$$. Is $$0.5x^2 - x + 0.25$$ a polynomial? Explain your thinking.

### Problem 3

What is the result when any 2 integers are multiplied?

A:

a positive integer

B:

a negative integer

C:

an integer

D:

an even number

### Problem 4

Clare wants to make an open-top box by cutting out corners of a 30 inch by 25 inch piece of poster board and then folding up the sides. 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 a reasonable domain for $$V$$ in this context?

### Solution

(From Unit 2, Lesson 1.)

### Problem 5

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

1. $$f(x)=2x^5 - 8 x^2 - x - 6$$
2. $$h(x)=x^3 - 7 x^2 - x + 2$$
3. $$g(x)=5 x^2-4 x^3 + 2x +5.4$$

### Solution

(From Unit 2, Lesson 3.)

### Problem 6

Which point is a relative minimum?

A:

A

B:

B

C:

C

D:

D