Question
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
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.
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.
Answers
GPT-4o mini
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**.
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**.