Question

Find the missing values in the factored form of q(x)
, given its graph.

(1 point)

Answer 1

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:

1. **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.

2. **Y-intercept:** Determine the point where the graph crosses the y-axis, which gives you the constant term when evaluating \( q(0) \).

3. **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.

4. **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.