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How do you factor and solve # r^3 + 3r^2 + 3r + 1 = 0#?

Answer 1

r = -1

#f(r) = r^3 + 3r^2 + 3r + 1 = 0# Use algebra identity: #(r + 1)^3 = r^3 + 3r^2 + 3r + 1 # #f(r) = (r + 1)^3# Answer: r = - 1

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Answer 2

Eli Assouline

To factor and solve the equation ( r^3 + 3r^2 + 3r + 1 = 0 ), you can use the technique of synthetic division or the rational root theorem. However, in this case, the rational root theorem suggests that the only possible rational roots are ±1, which do not satisfy the equation. Therefore, we can conclude that the roots are irrational or complex. By observing the equation, we can recognize that it is a perfect cube, specifically ((r + 1)^3). Thus, the equation can be rewritten as ((r + 1)^3 = 0). Solving for ( r ) yields ( r = -1 ). Therefore, the only real root of the equation is ( r = -1 ).

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.