Given the function f(x)=x^4+10x^3+35x^2+50x+24 , factor completely. Show all work and steps. Then sketch the graph.
2 answers:
I would start by graphing this using a graphing calculator.
You have correctly found the factoring.
About the graph:
The function is 4th degree, so the overall shape will be that of a U, extending to +∞ for both positive and negative values of x.
Since there are 4 real roots, the graph crosses the x-axis 4 times, and you know those crossings are at x = -4, -3, -2, -1. Because of the overall shape, you know these wiggles will make the curve look like a W. Because the roots are close together, the difference between peak and valleys will not be great. This is enough to get you the general shape of the graph.
Since the roots are symmetrical about x=-2.5, that will be the location of the relative maximum. Its value can be found to be 9/16, by evaluating the f(x) for that value of x. The locations of the relative minima can be found by factoring (2x+5) from the derivative cubic. This gives you a quadratic with roots at -2.5±√1.25, the locations of the relative minima. Evaluating f(x) at these points will tell you their values (-1).
So, your graph comes screaming down from +∞ to make a zero crossing at x=-4 to a minimum of -1 at about x=-3.62, then crosses zero again at x=-3 to a maximum of 0.5625 at x=-2.5. The curve is mirrored about the line of symmetry at x=-2.5 to give you two more zero crossings at x=-2 and x=-1 with another minimum of -1 at about x=-1.38. After crossing zero at x=-1, it crosses the y-axis at f(0) = 24. (Due to symmetry, you know that (-5, 24) is also a point on the graph.)
You have correctly found the factoring.
About the graph:
The function is 4th degree, so the overall shape will be that of a U, extending to +∞ for both positive and negative values of x.
Since there are 4 real roots, the graph crosses the x-axis 4 times, and you know those crossings are at x = -4, -3, -2, -1. Because of the overall shape, you know these wiggles will make the curve look like a W. Because the roots are close together, the difference between peak and valleys will not be great. This is enough to get you the general shape of the graph.
Since the roots are symmetrical about x=-2.5, that will be the location of the relative maximum. Its value can be found to be 9/16, by evaluating the f(x) for that value of x. The locations of the relative minima can be found by factoring (2x+5) from the derivative cubic. This gives you a quadratic with roots at -2.5±√1.25, the locations of the relative minima. Evaluating f(x) at these points will tell you their values (-1).
So, your graph comes screaming down from +∞ to make a zero crossing at x=-4 to a minimum of -1 at about x=-3.62, then crosses zero again at x=-3 to a maximum of 0.5625 at x=-2.5. The curve is mirrored about the line of symmetry at x=-2.5 to give you two more zero crossings at x=-2 and x=-1 with another minimum of -1 at about x=-1.38. After crossing zero at x=-1, it crosses the y-axis at f(0) = 24. (Due to symmetry, you know that (-5, 24) is also a point on the graph.)
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