Question

what is the degree and sign of the leading coefficient for this polynomial?
options:
odd degree + leading coeff.
odd degree - leading coeff.
even degree + leading coeff.
even degree - leading coeff.

Explanation:

Step1: Analyze End Behavior (Degree)

For a polynomial, the end behavior (what happens as \( x \to \pm\infty \)) depends on the degree (even or odd) and the leading coefficient's sign. If the ends of the graph go in opposite directions (one up, one down), the degree is odd. If both ends go in the same direction, the degree is even. Here, as \( x \to +\infty \), the graph goes down, and as \( x \to -\infty \), the graph goes down? Wait, no—wait, looking at the graph: left end (as \( x \to -\infty \)): the graph goes down (since the arrow is pointing down), right end (as \( x \to +\infty \)): the graph also goes down? Wait, no, wait the left end: when \( x \) approaches \( -\infty \), the graph's left end (the part for very negative \( x \))—wait, the graph's left side: the arrow is pointing down (so as \( x \to -\infty \), \( y \to -\infty \))? Wait, no, the left end: the graph starts at the left (x=-2 area) with an arrow pointing down? Wait, no, the standard end behavior: for odd degree, ends go in opposite directions; for even, same. Wait, looking at the graph: the right end (as \( x \to +\infty \)): the graph is going down (arrow pointing down), and the left end (as \( x \to -\infty \)): the graph is going down? Wait, no, that can't be. Wait, maybe I misread. Wait, the left end: when \( x \) is very negative, the graph's leftmost part—if the leading term is \( a_n x^n \), for odd \( n \), if \( a_n > 0 \), then as \( x \to +\infty \), \( y \to +\infty \), and \( x \to -\infty \), \( y \to -\infty \); if \( a_n < 0 \), then \( x \to +\infty \), \( y \to -\infty \), \( x \to -\infty \), \( y \to +\infty \). For even \( n \), if \( a_n > 0 \), both ends up; if \( a_n < 0 \), both ends down. Wait, in the graph, both ends are going down (left end: as \( x \to -\infty \), \( y \to -\infty \); right end: as \( x \to +\infty \), \( y \to -\infty \))? Wait, no, that would be even degree with negative leading coefficient. Wait, but the options: let's check the options. The options have "Even Degree - Leading Coeff." (teal), "Even Degree + Leading Coeff." (orange), "Odd Degree - Leading Coeff." (purple), "Odd Degree + Leading Coeff." (green). Wait, no—wait, the right end: as \( x \to +\infty \), the graph is decreasing (going down), and the left end: as \( x \to -\infty \), the graph is decreasing? Wait, no, that's even degree. Wait, no, wait: if both ends go down, that's even degree with negative leading coefficient. Wait, let's re-express:

• Even degree: ends same direction.
• Odd degree: ends opposite.

So in the graph, both ends (left and right) are going down (since the arrows on both ends are pointing down). So that's even degree. Now, the sign of leading coefficient: for even degree, if leading coefficient is positive, both ends go up; if negative, both go down. So since both ends go down, leading coefficient is negative. So degree is even, leading coefficient is negative.

Step2: Confirm with Options

The options: the teal one is "Even Degree - Leading Coeff.", which matches. The purple is "Odd Degree - Leading Coeff." (ends opposite, but here ends same, so odd is wrong). Green: odd + (ends opposite, but here ends same, and + would have ends opposite with + leading coeff for odd: no). Orange: even + (ends up, but here ends down). So the correct one is Even Degree, negative leading coefficient.

Answer:

Even Degree - Leading Coeff.