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Its readers span a broad spectrum of mathematical interests, and include professional mathematicians as well as students of mathematics at all collegiate levels. HenceĬoefficient e is found using the y intercept (0, -2) from the graph.The Monthly publishes articles, as well as notes and other features, about mathematics and the profession. The terms b x 3 and d x included in the given expression of the polynomial above are not even and therefore their coefficients are equal to 0. The graph of the polynomial is symmetric with respect to the y axis and therefore the polynomial function given above must be an even function. The graph of polynomial $$y=a x^4+bx^3+c x^2+d x+e$$ is shown below. We now compare the expression of the polynomial found above toĪnd obtain the values of the coefficients We now expand the polynomial, write it in standard form and identify the coefficient a, b, c and d. We now need to find k using the y -intercept (0, 1) shown in the graph. The graph of the polynomial has a zero of multiplicity 1 at x = -2 which corresponds to the factor x + 2 and a zero of multiplicity 2 at x = 1 which corresponds to the factor (x - 1) 2. The graph of a cubic polynomial $$ y = a x^3 + b x^2 +c x + d $$ is shown below. The equation of polynomial f(x) is given by.
Using the zeros at x = 0 and x = 5 / 2, f(x) may be written asį(x) = k (x - 0) 3 (x - 5 / 2), where k is a constant. The shape of the graph at x = 1/2 is close to linear hence the zero at x = 5 / 2 has multiplicity equal to 1. The graph at x = 0 has an 'cubic' shape and therefore the zero at x = 0 has multiplicity of 3. Because the graph crosses the x axis at x = 0 and x = 5 / 2, both zero have an odd multiplicity. These x intercepts are the zeros of polynomial f(x). The graph has x intercepts at x = 0 and x = 5 / 2. The graph of the polynomial has a zero of multiplicity 1 at x = 2 which corresponds to the factor (x - 2), another zero of multiplicity 1 at x = -2 which corresponds to the factor (x + 2), and a zero of multiplicity 2 at x = -1 (graph touches but do not cut the x axis) which corresponds to the factor (x + 1) 2, hence the polynomial f has the equation:į(x) = k (x - 2)(x + 2)(x + 1) 2, where k is a constant.Ĭonstant k may be found using the y intercept f(0) = - 1 shown in the graph.įind the equation of the degree 4 polynomial f graphed below. G(x) = k (x + 1)(x - 3) 2, where k is a constant.Ĭonstant k may be found using the point with coordinates (1, 3) shown in the graph.įind the fourth-degree polynomial function f whose graph is shown in the figure below. The graph of the function has one zero of multiplicity 1 at x = -1 which corresponds to the factor x + 1 and and a zero of multiplicity 2 at x = 3 (graph touches but do not cut the x axis) which corresponds to the factor (x - 3) 2, hence function g has the equation:
How to find a polynomial given its graph? Questions are presented along with their detailed solutions and explanations.įind the equation of the cubic polynomial function g shown below.