Consider the graph of the polynomial function p(x). Given that (2x+3) is a factor of p(x), which of the following could be the multiplicity of (2x+3)?
(1 point)
−1.5
1
1.5
2
The multiplicity of a factor in a polynomial function represents the number of times that factor is repeated.
If (2x+3) is a factor of p(x), it means that when we divide p(x) by (2x+3), the remainder is 0.
Let's consider each option:
-1.5: The multiplicity cannot be negative or a decimal number.
1: The multiplicity can be 1. If the factor (2x+3) appears once in the function, it means that there is one root at x = -3/2.
1.5: The multiplicity cannot be a decimal number.
2: The multiplicity can also be 2. If the factor (2x+3) appears twice in the function, it means that there is a double root at x = -3/2.
So, the possible multiplicities for (2x+3) are 1 and 2. The correct option is 2.
To determine the possible multiplicities, we need to use the Factor Theorem. The Factor Theorem states that if (2x+3) is a factor of p(x), then p(-3/2) = 0.
Let's substitute (-3/2) into the polynomial p(x) and see if it equals zero. If it does, then the multiplicity of (2x+3) is 1. Otherwise, we will need to consider other possible multiplicities.
So, evaluate p(-3/2) by substituting (-3/2) in place of x in the polynomial p(x). Then check if the result is equal to zero.
By multiplying (2x+3) by its conjugate (2x-3), we obtain:
(2x+3)(2x-3) = 4x^2 - 9
So, (2x+3) is a factor of p(x) if and only if (4x^2 - 9) is divisible by (2x+3).
To check if (4x^2 - 9) is divisible by (2x+3), divide (4x^2 - 9) by (2x+3) using long division or synthetic division:
2x - 1
_______________
2x + 3 | 4x^2 + 0x - 9
- (4x^2 + 6x)
_______________
-6x - 9
+ (6x + 9)
_______________
0
Since the remainder is zero, we can conclude that (4x^2 - 9) is divisible by (2x+3) and therefore, (2x+3) is a factor of p(x).
Therefore, the multiplicity of (2x+3) could be 1.
To determine the possible multiplicity of (2x+3) as a factor of p(x), we need to refer to the graph of p(x). Unfortunately, we don't have access to the graph in this text-based format.
However, we can use the concept of multiplicity to determine the possible values. The multiplicity of a factor determines how many times it appears in the polynomial.
Since (2x+3) is a linear factor, the possible multiplicities are positive integers. This means that the multiplicity can be 1, 2, 3, and so on.
Therefore, among the options given, the correct answer would be 1 or 2.