# Lesson 23

Polynomial Identities (Part 1)

### Problem 1

Select all the identities:​​​​

A:

$$(x+2)^3 = x^3 + 8$$

B:

$$(x^6 + x) = (x-1)(x^5 + x^4 + x^3 + x^2 + x)$$

C:

$$(x^2 - 1)(x^4 + x^2 + 1) = x^6 - 1$$

D:

$$(x+1)^4 = x^4 + x^3 + x^2 + x + 1$$

E:

$$(x+1)(x^4 - x^3 + x^2 - x + 1) = x^5+1$$

F:

$$(x^3 - 1)(x^3 + 1) = x^6 - 1$$

### Solution

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### Problem 2

Is $$2(x+1)^2 = (2x+2)^2$$ an identity? Explain or show your reasoning.

### Solution

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### Problem 3

Mai is solving the rational equation $$5 = \frac{2+7x}{x}$$ for $$x$$. What move do you think Mai would make first to solve for $$x$$? Explain your reasoning.

### Solution

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### Problem 4

For $$x$$-values of 0 and -2, $$(x^5+32) = (x+2)^5$$. Does this mean the equation is an identity? Explain your reasoning.

### Solution

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### Problem 5

Clare finds an expression for $$S(r)$$ that gives the surface area in square inches of any cylindrical can with a specific fixed volume, in terms of its radius $$r$$ in centimeters. This is the graph Clare gets if she allows $$r$$ to take on any value between -1.2 and 3.

1. What would be a more appropriate domain for Clare to use instead?
2. What is the approximate minimum surface area for her can?

### Solution

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(From Unit 2, Lesson 16.)

### Problem 6

Which values of $$x$$ make $$\frac{3x+1}{x}=\frac{1}{x-3}$$ true?

### Solution

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(From Unit 2, Lesson 22.)