To find the missing values in the factored form of a polynomial \( q(x) \) using its graph, you typically need to look for the following key features:
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Roots (x-intercepts): Identify the points where the graph crosses or touches the x-axis. Each x-intercept represents a root of the polynomial. If the graph just touches the x-axis and does not cross it, the corresponding factor is squared.
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Y-intercept: Determine the point where the graph crosses the y-axis, which gives you the constant term when evaluating \( q(0) \).
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End Behavior: Observing how the graph behaves as \( x \) approaches positive or negative infinity can help determine the leading coefficient (positive or negative) and the degree of the polynomial.
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Multiplicity of roots: If a root appears more than once (the graph touches but does not cross the x-axis), that root's factor will have an exponent greater than one in the factored form.
Once you gather this information, you can write the polynomial in factored form as:
\[ q(x) = a(x - r_1)^{m_1}(x - r_2)^{m_2} \ldots (x - r_n)^{m_n} \]
where \( r_i \) are the roots, \( m_i \) are their respective multiplicities, and \( a \) is a leading coefficient based on the graph's end behavior.
If you can provide more specifics about the graph (e.g., coordinates of key points, the number and nature of x-intercepts), I can assist you further in determining the exact factored form.
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