Guided by the journal of a vanished expedition, Emek and Tara explored an abandoned canyon marked with carved symbolic clues. Each clue revealed an algebraic expression that had to be simplified before the journal disclosed the next location. As they progressed, the expressions became more complex, requiring them to expand products, factor expressions, combine like terms, and verify equivalent forms before continuing. They carefully checked every simplification to avoid following false trails. After solving the final cipher, they reached the expedition's last campsite, where they uncovered preserved scientific records and rare geological samples that explained the explorers' remarkable discoveries.
Challenges
Challenge 1 of 5
Challenges
Challenge 1 of 4
Hint
Multiply every term in the first factor by every term in the second factor.
A carved clue shows (2x - 3)(x + 4). Tara claims it expands to 2x^2 + 5x - 12. Which statement correctly verifies her work?
Tara is correct because the expanded expression is 2x^2 + 5x - 12.Tara is incorrect because the expanded expression is 2x^2 + 11x - 12.Tara is incorrect because the expanded expression is 2x^2 - 5x - 12.Tara is correct because only the first and last terms need to be multiplied.Correct! Expanding and combining like terms gives 2x^2 + 5x - 12.
Check all four products before combining the two middle terms.
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Hint
Evaluate the power first, then multiply and subtract.
The next cipher displays 3(x - 2)^2 - 2(x + 1). What is its value when x = 4?
Correct! The cipher has a value of 2 when x = 4.
Substitute 4 carefully and follow the order of operations.
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Hint
First factor out x, then factor the remaining difference of squares.
Emek must choose an expression equivalent to x^3 - 4x. Which option correctly factors the cipher completely?
x(x - 4)x(x - 2)^2x(x - 2)(x + 2)(x - 4)(x + 1)Correct! x^3 - 4x factors completely as x(x - 2)(x + 2).
Factor out the greatest common factor before using the difference of squares identity.
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