Question

what is the exponent?

(y = (x + 4)(x + 3)^{?}(x + 1))

3
2
1

Explanation:

Identify the root of interest

We analyze the given polynomial function:

$$y = (x+4)(x+3)^{[?]}(x+1)$$

The exponent we need to find is associated with the factor \((x+3)\), which corresponds to the root \(x = -3\).

Analyze graph behavior at the root

We observe the behavior of the graph at \(x = -3\), which is highlighted by the red circle.
At \(x = -3\), the graph touches the x-axis and turns around (tangent to the x-axis) without crossing it.

Determine the multiplicity

A root where the graph touches and turns around has an even multiplicity.
Among the given choices (3, 2, 1), the only even integer is 2.
Thus, the exponent must be 2.

Answer:

• 3
• 2 (Correct answer)
• 1