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Sketch a graph of a fifth-degree polynomial f(x) such that f(x) < 0 when x< -2 or 4 < x < 7, and f(x)>0 when -2<x<0, 0<x<4 or when x>7. What conclusions can you make about the sign of the leading coefficient and its end behaviour (the sign changes the direction of the graph)? Explain your answer using terminology

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Did you know?

Did you know that the graph of a fifth-degree polynomial can have distinct sections where it is either above or below the x-axis? This means that there can be intervals where the polynomial is negative and others where it is positive. In this particular case, the given conditions tell us that the polynomial is negative for x values less than -2 and between 4 and 7, and positive for x values between -2 and 0, 0 and 4, and greater than 7. Based on these conditions, we can conclude that the leading coefficient of the polynomial must be positive. This is because a positive leading coefficient will cause the graph to start in the negative y-values and end in positive y-values as x approaches positive or negative infinity. The end behavior refers to how the graph behaves as x approaches very large or very small values. Since the graph changes its sign, we can say that the graph will have both a positive and negative y-intercept, meaning the polynomial crosses the x-axis at least three times.