Use the equation of the polynomial function ​f(x)equalsleft parenthesis x minus 1 right parenthesis left parenthesis x squared minus 4 right parenthesis to complete the follow

Question

The curve opens
▼
up
down
to the right because the leading coefficient is
▼
positive
negative
. Because the polynomial is
▼
cubic,
quartic,
the graph has end behaviors in the
▼
opposite
same
​direction, so the other end opens
▼
up
down
to the left.

Answer 1

To analyze the polynomial function \( f(x) = (x - 1)(x^2 - 4) \), we can first expand it:

1. The term \( x^2 - 4 \) can be factored as \( (x - 2)(x + 2) \).
2. Therefore, we can rewrite \( f(x) \) as: \[ f(x) = (x - 1)(x - 2)(x + 2) \]

Now, we recognize that this is a polynomial of degree 3, which is cubic.

Next, let's analyze the leading coefficient:

1. The leading term after expanding this polynomial will be \( x^3 \) since multiplying the leading terms of each factor yields \( x \cdot x \cdot x = x^3 \).
2. The leading coefficient is 1, which is positive.

Now we can summarize the findings:

• The curve opens up because the leading coefficient is positive.
• Because the polynomial is cubic, the graph has end behaviors in the opposite direction, so the other end opens down.

Putting this all together, here are the completed statements:

The curve opens up because the leading coefficient is positive. Because the polynomial is cubic, the graph has end behaviors in the opposite direction, so the other end opens down.