The polynomial function is of degree n. The sum of the multiplicities must be n. Starting from the left, the first zero occurs at [latex]x=-3\\[/latex]. 4 + 6i, -2 - 11i -1/3, 4 + 6i, 2 + 11i -4 + 6i, 2 - 11i 3, 4 + 6i, -2 - 11i Can I have some guidance Precalculus Write a polynomial function of minimum degree in standard form with real coefficients whose zeros and their multiplicities include those listed. Keep this in mind: Any odd-multiplicity zero that flexes at the crossing point, like this graph did at x = 5, is of odd multiplicity 3 or more. We call this a triple zero, or a zero with multiplicity 3. Therefore the zero of the quadratic function y = x^{2} is x = 0. Multiplicity is how many times a certain solution to the function. The zero associated with this factor, [latex]x=2[/latex], has multiplicity 2 because the factor [latex]\left(x - 2\right)[/latex] occurs twice. Descartes also tells us the total multiplicity of negative real zeros is 3, which forces -1 to be a zero of multiplicity 2 and - \frac {\sqrt {6}} {2} to have multiplicity 1. The sum of the multiplicities is the degree. The x-intercept [latex]x=-1[/latex] is the repeated solution of factor [latex]{\left(x+1\right)}^{3}=0[/latex]. Sometimes the graph will cross over the x-axis at an intercept. The sum of the multiplicities must be 6. The graph passes through the axis at the intercept, but flattens out a bit first. The multiplicity of a root is the number of times the root appears. The factor is quadratic (degree 2), so the behavior near the intercept is like that of a quadratic—it bounces off of the horizontal axis at the intercept. It just "taps" it, … The last zero occurs at [latex]x=4[/latex]. The x-intercept [latex]x=2\\[/latex] is the repeated solution of equation [latex]{\left(x - 2\right)}^{2}=0\\[/latex]. The factor is repeated, that is, the factor [latex]\left(x - 2\right)\\[/latex] appears twice. \[ \begin{align*} 2x+1=0 \\[4pt] x &=−\dfrac{1}{2} \end{align*}\] The zeros of the function are 1 and \(−\frac{1}{2}\) with multiplicity 2… For higher odd powers, such as 5, 7, and 9, the graph will still cross through the x-axis, but for each increasing odd power, the graph will appear flatter as it approaches and leaves the x-axis. Don't forget the multiplicity of x, even if it doesn't have an exponent in plain view. The zero associated with this factor, [latex]x=2[/latex], has multiplicity 2 because the factor [latex]\left(x - 2\right)[/latex] occurs twice. Use the graph of the function of degree 6 to identify the zeros of the function and their possible multiplicities. This is a single zero of multiplicity 1. The factor theorem states that is a zero of a polynomial if and only if is a factor of that polynomial, i.e. This is called multiplicity. For more math shorts go to www.MathByFives.com It means that x=3 is a zero of multiplicity 2, and x=1 is a zero of multiplicity 1. 232. We have two unique zeros: #-2# and #4#. The table below summarizes all four cases. Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share … For example, the polynomial P(x) = (x - 2)^237 has precisely one root, the number 2. The zero of –3 has multiplicity 2. The graph passes directly through the x-intercept at [latex]x=-3[/latex]. If a polynomial contains a factor of the form [latex]{\left(x-h\right)}^{p}[/latex], the behavior near the x-intercept h is determined by the power p. We say that [latex]x=h[/latex] is a zero of multiplicity p. The graph of a polynomial function will touch the x-axis at zeros with even multiplicities. 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