Lesson 23

Select all the identities:​​​​

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

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

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

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

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

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

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

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.

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

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?

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