Polynomial Functions, Zeros, And Factorization: A Comprehensive Analysis

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Polynomial Functions, Zeros, and Factorization: A Comprehensive Analysis 1. Given that the function f(x)f(x)f(x) has vertical asymptotes at x=4x = 4x=4 and x=−1x = - 1x=−1, and that the function equals zero at x=1x = 1x=1 and x=7x = 7x=7, which of the following represents the denominator of the function? Possible Answers: A) (x−4) ⋅ (x+1)(x - 4) \ cdot (x + 1)(x−4) ⋅ (x+1) B) (x−4)2 ⋅ (x+1)2(x - 4)^2 \ cdot (x + 1)^2(x−4)2 ⋅ (x+1)2 C) (x−4)3 ⋅ (x+1)3(x - 4)^3 \ cdot (x + 1)^3(x−4)3 ⋅ (x+1)3 D) (x−4) ⋅ (x+1)3(x - 4) \ cdot (x + 1)^3(x−4) ⋅ (x+1)3 Answer: This function has two vertical asymptotes x = 4 and x = - 1 hence the denominator is in the form : ( 𝑥 − 4 ) 𝛼 ( 𝑥 + 1 ) 𝛽 And f(x) = 0 when x = 1 and x = 7 Therefore by looking at the solutions the only possible answer is B (where α = 2 and β = 2) 2. Given that the function f(x)f(x)f(x) has vertical asymptotes at x=−3x = - 3x=−3 and x=3x = 3x=3, and the denominator is in the form x2−9x^2 - 9x2−9, which of the following describes the correct numerator of the function in order to achieve positive values of f(x)f(x)f(x) between (−4,4)( - 4, 4)(−4,4) and negative values otherwise? Possible Answers: A) x2x^2x2 B) −x2 - x^2−x2 C) x3x^3x3 D) −x3 - x^3−x3 Answer: This function has two vertical asymptotes x = - 3 and x = 3 hence the denominator is in the form : 𝑥 2 − 9 In order to have the given graph with positive values of f between ( - 4,4) and negative values otherwise the numerator has to be – x² Therefore the answer is B 3. Solve the inequality (x−2)2(x+6)<0(x - 2)^2(x+6) < 0(x−2)2(x+6)<0.
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