the function f is given by ( f(x)=\frac{5 x^{3}-9 x + 4}{x^{2}-3 x + 2} ). at which of the following values…

the function f is given by ( f(x)=\frac{5 x^{3}-9 x + 4}{x^{2}-3 x + 2} ). at which of the following values of x does the graph of f have an x - intercept?\na ( x = 1 ) and ( x = 2 ) only\nb ( x=-1.525 ) and ( x = 0.525 ) only\nc ( x=-1.525, x = 0.525 ), and ( x = 2 ) only\nd ( x=-1.525, x = 0.525, x = 1 ), and ( x = 2 )

the function f is given by ( f(x)=\frac{5 x^{3}-9 x + 4}{x^{2}-3 x + 2} ). at which of the following values of x does the graph of f have an x - intercept?\na ( x = 1 ) and ( x = 2 ) only\nb ( x=-1.525 ) and ( x = 0.525 ) only\nc ( x=-1.525, x = 0.525 ), and ( x = 2 ) only\nd ( x=-1.525, x = 0.525, x = 1 ), and ( x = 2 )

Answer

Explanation:

Step1: Recall the definition of x - intercept

The x - intercept of a function (y = f(x)) occurs when (y=f(x)=0). For a rational function (f(x)=\frac{N(x)}{D(x)}), (f(x) = 0) when (N(x)=0) (and (D(x)\neq0)).

Step2: Set the numerator equal to zero

Set (N(x)=5x^{3}-9x + 4=0). We can try to find a root of the polynomial (5x^{3}-9x + 4) by trial - and - error. Let's test (x = 1): (N(1)=5\times1^{3}-9\times1 + 4=5 - 9 + 4=0). Since (x = 1) is a root of (5x^{3}-9x + 4), we can factor (5x^{3}-9x + 4=(x - 1)(ax^{2}+bx + c)). Using polynomial long - division or synthetic division: (5x^{3}-9x + 4=(x - 1)(5x^{2}+5x - 4)). Then, use the quadratic formula (x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}) for (ax^{2}+bx + c = 5x^{2}+5x - 4), where (a = 5), (b = 5), (c=-4). (x=\frac{-5\pm\sqrt{5^{2}-4\times5\times(-4)}}{2\times5}=\frac{-5\pm\sqrt{25 + 80}}{10}=\frac{-5\pm\sqrt{105}}{10}\approx\frac{-5\pm10.25}{10}). (x_1=\frac{-5 + 10.25}{10}=0.525) and (x_2=\frac{-5-10.25}{10}=-1.525).

Step3: Check the denominator

Set (D(x)=x^{2}-3x + 2=(x - 1)(x - 2)). The function (f(x)) is undefined when (x = 1) or (x = 2) (because (D(1)=0) and (D(2)=0)).

Answer:

B. (x=-1.525) and (x = 0.525) only