find the domain, x intercepts and y intercepts of fourth root x^(5) (x - 4)^2.

Answer 1

y-intercept: A y-intercept occurs where the graph crosses the y-axis, which means its x-coordinate needs to be 0, so we will plug 0 into the expression to find it; (0^5)(0-4)^2=0*4=0, hence the y-intercept is 0 or (0,0).

x-intercept: An x-intercept occurs where the graph intersects the x-axis, which occurs when the y-coordinate is 0, so we set the expression equal to 0 and solve for x; 0=(x^5)(x-4)^2, since we have a product equalling 0, the solutions occur when each factor equals 0, hence, x^5=0 or (x-4)^2=0, which implies x=0 or x-4=0 which implies the x-intercepts are 0 or 4 or the or in coordinate form (0,0) and (4,0).

Domain: We realize by foiling the second term in the expression and then distributing the first, that this is really just a polynomial, (x^5)(x-4)^2=(x^5)(x^2-8x+16)=x^7-8x^6+16x^5. The domain for all polynomials is all real numbers or (-infinity, infinity).