Did you know?
Did you know that synthetic division can be used to find the zeroes of polynomial functions? For example, to find the zeroes of the function f(x)=2x^3+x^2-5x+2, you can use synthetic division. This method allows you to divide the polynomial by factors of the form (x-a), where a is a possible zero. By trying different values of a and using synthetic division, you can determine the zeroes of the function.
Additionally, did you know that for the function f(x)=5(x+4)/x^2+x-12, you can analyze various properties?
(a) The domain of this function would be all real numbers except for the values of x that make the denominator zero, which is x^2. Therefore, the domain is all real numbers except x=0.
(b) To find the intercepts, set x=0 to find the y-intercept, which is (0, 0). To find the x-intercepts, set y=0 and solve the equation 5(x+4)/x^2+x-12=0.
(c) To determine vertical or horizontal asymptotes, a vertical asymptote occurs when the denominator approaches zero, but in this case, the denominator can never be equal to zero. Therefore, there are no vertical asymptotes. As for horizontal asymptotes, you can analyze the degree of the numerator and the denominator. Since the degree of the numerator is 1 and the degree of the denominator is 2, there will be a horizontal asymptote at y=0.
(d) Finally, to identify any removable discontinuities, you would need to check if there are any common factors between the numerator and the denominator. In this case, there are no common factors, so there are no removable discontinuities.
