# 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.

### Solution

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### 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.

### Solution

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### Problem 3

What is the result when any 2 integers are multiplied?

a positive integer

a negative integer

an integer

an even number

### Solution

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### 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.

- Write an expression for \(V(x)\).
- What is a reasonable domain for \(V\) in this context?

### Solution

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(From Unit 2, Lesson 1.)### Problem 5

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

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

### Solution

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(From Unit 2, Lesson 3.)### Problem 6

Which point is a relative minimum?

A

B

C

D

### Solution

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(From Unit 2, Lesson 3.)