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)
Responses
−1.5
negative 1.5
1.5
1.5
2
2
1
The correct answer is:
2
To determine the multiplicity of the factor (2x + 3), we need to look at how many times the factor appears in the factorization of the polynomial function p(x).
If (2x + 3) is a factor of p(x), it means that when we divide p(x) by (2x + 3), the remainder is zero.
The multiplicity of a factor can be found by examining the behavior of the graph at the corresponding x-intercept. If the graph crosses the x-axis at the x-intercept, the multiplicity is odd. If the graph touches (but does not cross) the x-axis at the x-intercept, the multiplicity is even.
In this case, since (2x + 3) is a factor of p(x), the graph of p(x) will cross the x-axis at the x-intercept (-3/2). This means the multiplicity of (2x + 3) is odd. Therefore, the possible multiplicity options are either -1.5 or 1.5.
So, the correct answer options are:
-1.5 or negative 1.5
1.5 or 1.5
To determine the multiplicity of a factor in a polynomial function, we need to consider the exponent of that factor in the function.
In this case, the given factor is (2x+3). If this factor is a factor of p(x), it means that when we divide p(x) by (2x+3), there is no remainder.
We can use synthetic division to check if (2x+3) is a factor of p(x) and determine the multiplicity. Here's how to do it:
1. Write the polynomial function p(x) in the form p(x) = (2x+3)q(x) + r(x), where q(x) is the quotient and r(x) is the remainder.
2. Set up the synthetic division table, with the factor (2x+3) as the divisor and the coefficients of p(x) as the dividend.
3. Perform synthetic division using the coefficients of p(x) and the factor (2x+3) to find the quotient.
4. If the remainder, r(x), is equal to zero, then (2x+3) is a factor of p(x). The multiplicity of (2x+3) is equal to the number of times (2x+3) divides evenly into p(x).
Let's apply this process to the given options:
- For option -1.5 or negative 1.5:
Since the multiplicity must be a non-negative integer, -1.5 cannot be the multiplicity for the factor (2x+3).
- For option 1.5:
Again, since the multiplicity must be a non-negative integer, 1.5 cannot be the multiplicity for the factor (2x+3).
- For option 2:
To determine if 2 could be the multiplicity, we need to divide p(x) by (2x+3). If the remainder is zero, then 2 could be the multiplicity.
- For option 1:
Similarly, to determine if 1 could be the multiplicity, we need to divide p(x) by (2x+3) and check if the remainder is zero.
By performing the synthetic division for both options 2 and 1, we can determine which one gives a remainder of zero. The value with a remainder of zero would be the possible multiplicity for the factor (2x+3).