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Polynomial Functions, Zeros, And Factorization: A Comprehensive Analysis

Understand polynomial functions with this Assignment Solution filled with examples.

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Polynomial Functions, Zeros, and Factorization: A Comprehensive Analysis1.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 zeroat x=1x = 1x=1 and x=7x = 7x=7, which of the followingrepresents 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)2C) (x−4)3⋅(x+1)3(x-4)^3\cdot (x + 1)^3(x−4)3⋅(x+1)3D) (x−4)⋅(x+1)3(x-4)\cdot (x + 1)^3(x−4)⋅(x+1)3Answer:This function has two vertical asymptotes x = 4 and x =-1 hence the denominator is in the form :(𝑥−4)𝛼(𝑥+1)𝛽And f(x) = 0 when x = 1 and x = 7Therefore by looking at the solutions the only possible answer is B (whereα = 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 thecorrect 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^2x2B) −x2-x^2−x2C) x3x^3x3D) −x3-x^3−x3Answer:This function has two vertical asymptotes x =-3 and x = 3 hence the denominator is in the form :𝑥2−9In order to have the given graph with positive values of f between (-4,4) and negative valuesotherwise the numerator has to be–x²Therefore the answer is B3.Solve theinequality (x−2)2(x+6)<0(x-2)^2(x+6) < 0(x−2)2(x+6)<0.

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