Student question, Grade 12 level A polynomial function is described by the following conditions: f(x) has two real zeros at x= 2 and x= 1, each of multiplicity 2 f(x) has two complex zeros as x goes to infinity, f(x) goes to +infinity as x goes to +infinity, f(x) goes to +infinity f(x) has a yintercept at (0,2) Sketch a polynomial function that satisfies the above conditions. Write an equation for function f(x) Write another function g(x) that also satisfies the above parameters. Thanks! Hi Where I am today I don't have a very good drawing tool so my graphs aren't very smooth.
Your polynomial has at least 6 zeros, a double zero of multiplicity 2 at 2, a zero of multiplicity 2 at 1 and 2 complex zeros. Thus the polynomial must have degree at least 6. "As x goes to infinity, f(x) goes to +infinity" implies that as x gets large and negative, y gets large and positive, so for large negative values of x the graph is in the second quadrant. This allows me to draw a rough sketch of what the graph might look like. Starting in the second quadrant draw a graph with enough peaks and valleys that it crosses the xaxis 6 times (6 times because it has 6 roots)
This sketch has the property that "as x goes to infinity, f(x) goes to +infinity" and "as x goes to +infinity, f(x) goes to +infinity" but the zero are not correct. A zero of multiplicity 2 results when a peak or valley just touches the xaxis as in the diagrams below. and complex zeros result when a peak or valley does not extend far enough for the curve to cross the xaxis. Hence the graph might look like To finish the sketch you need to put the yaxis somewhere so that the graph passes through (0,2). For the algebra part of the problem you know that f(x) is a polynomial of degree 6. You also know that it has "a double zero of multiplicity 2 at 2" and "a zero of multiplicity 2 at 1". Hence you can factor f(x) and get the 4 factors Can you find a, b and c so that all the conditions are satisfied? Peny
