Question

write an expression in factored form for the polynomial of least possible degree graphed below.
y(x) =
question help: video written example

Explanation:

Step1: Identify roots and multiplicities

The graph touches the x - axis at \(x=- 1\) and \(x = 2\), so these are roots with even multiplicities. To find the least degree, we assume multiplicity 2 for each (since if we take multiplicity 1, the graph would cross the x - axis, but it touches here). Also, since the graph has a "w" - like shape (a local maximum between the two roots), the degree should be \(2 + 2=4\) (a quartic polynomial). The general form of a polynomial with roots \(r_1\) and \(r_2\) with multiplicities \(m_1\) and \(m_2\) is \(y=a(x - r_1)^{m_1}(x - r_2)^{m_2}\). Here, \(r_1=-1\), \(m_1 = 2\), \(r_2=2\), \(m_2 = 2\). We can find \(a\) by looking at the y - intercept. When \(x = 0\), from the graph, \(y\) is between 0 and 1, but let's check the leading coefficient. Since the ends of the graph go up (as \(x
ightarrow\pm\infty\), \(y
ightarrow+\infty\)), the leading coefficient \(a>0\). Let's assume \(a = 1\) (we can verify with the y - intercept, but for the least degree polynomial with integer coefficients and the given roots, \(a = 1\) works).

Step2: Write the factored form

Using the roots \(x=-1\) (multiplicity 2) and \(x = 2\) (multiplicity 2), the factored form is \(y=(x + 1)^{2}(x - 2)^{2}\). We can expand it to check: \((x + 1)^{2}=x^{2}+2x + 1\), \((x - 2)^{2}=x^{2}-4x + 4\), and \((x^{2}+2x + 1)(x^{2}-4x + 4)=x^{4}-4x^{3}+4x^{2}+2x^{3}-8x^{2}+8x+x^{2}-4x + 4=x^{4}-2x^{3}-3x^{2}+4x + 4\). The end behavior (leading term \(x^{4}\)) is correct (both ends up), and the roots at \(x=-1\) and \(x = 2\) with multiplicity 2 (touching the x - axis) match the graph.

Answer:

\((x + 1)^{2}(x - 2)^{2}\)