Lesson 4

Combining Polynomials

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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:

a positive integer

B:

a negative integer

C:

an integer

D:

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.

  1. Write an expression for \(V(x)\).
  2. 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.

  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

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(From Unit 2, Lesson 3.)

Problem 6

Which point is a relative minimum?

graph with points A, B, C and D plotted. A is a relative minimum. B is on a decreasing portion, C is on a increasing portion, D is a relative maximum
A:

A

B:

B

C:

C

D:

D

Solution

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(From Unit 2, Lesson 3.)