Explanation The given expression is x 3 + 6 x 2 + 8 x 3 − x . Since the above radical represents a real number the fraction under the radical is greater than or equal to zero. Solve the inequality x 3 + 6 x 2 + 8 x 3 − x ≥ 0 as follows. Factor and simplify the above inequality as follows. x ( x 2 + 6 x + 8 ) 3 − x ≥ 0 x ( x ( x + 2 ) + 4 ( x + 2 ) ) 3 − x ≥ 0 x ( x + 4 ) ( x + 2 ) 3 − x ≥ 0 Setting each factors of numerator equal to zero the critical values are found as follows. ( x + 4 ) = 0 implies x = − 4 , ( x + 2 ) = 0 implies x = − 2 and x = 0 Again setting each factor of denominator equal to zero the critical values are found as follows. ( 3 − x ) = 0 implies x = 3 The critical values are x = 0 , x = − 4 , x = − 2 and x = 3 . Now, determine the sign of the given function for each of the intervals x < − 4 , − 4 < x < − 2 , − 2 < x < 0 , 0 < x < 3 and x > 3

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Chapter 17.3, Problem 34E

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The value of x for which the given radical represents the real number.

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Chapter 17.3, Problem 33E

Chapter 17.3, Problem 35E

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