Question

question 6.
molly graphed the function $f(x) = (x - 1)^2(x + 2)^3$. mollys graph shows that as $x \to -\infty$, $y \to -\infty$ and as $x \to \infty$, $y \to \infty$. is mollys graph correct? justify your answer.
a. \\(\boldsymbol{\circ}\\) molly is correct because the function represents an odd degree with a positive leading coefficient.
b. \\(\boldsymbol{\circ}\\) molly is correct because the function has a sum and a difference as factors.
c. \\(\boldsymbol{\circ}\\) molly is incorrect because the function has an even degree and an odd degree as factors.
d. \\(\boldsymbol{\circ}\\) it cannot be determined without looking at the graph.

Explanation:

To determine if Molly's graph is correct, we analyze the function \( f(x) = (x - 1)^2(x + 2)^3 \). First, find the degree of the polynomial by adding the exponents of the factors: \( 2 + 3 = 5 \), which is odd. The leading coefficient is the product of the leading coefficients of each factor. The leading term of \( (x - 1)^2 \) is \( x^2 \) (leading coefficient 1) and of \( (x + 2)^3 \) is \( x^3 \) (leading coefficient 1), so the leading coefficient of \( f(x) \) is \( 1 \times 1 = 1 \) (positive). For a polynomial with odd degree and positive leading coefficient, as \( x \to -\infty \), \( y \to -\infty \) and as \( x \to \infty \), \( y \to \infty \), which matches Molly's graph. Option A correctly justifies this by referencing the odd degree and positive leading coefficient. Option B's reasoning about sum and difference factors is irrelevant. Option C's reasoning about even and odd degree factors is incorrect (we consider the total degree). Option D is wrong as we can determine this from the polynomial's degree and leading coefficient.

Answer:

A. Molly is correct because the function represents an odd degree with a positive leading coefficient.