The 5th differences of a 5th degree polynomials are a constant value. (1 point)
a. True
b. False
b. False
a. True
To determine if the statement is true or false, we need to understand what is meant by the "5th differences" of a 5th-degree polynomial.
The differences of a polynomial are obtained by subtracting consecutive terms. For example, if we have a polynomial sequence 2, 6, 12, 20, 30, the first differences would be 6-2=4, 12-6=6, 20-12=8, 30-20=10.
The 5th differences, as the name suggests, refer to the differences between the 5th degree terms of a polynomial sequence. In other words, if we have a 5th-degree polynomial expressed as ax^5 + bx^4 + cx^3 + dx^2 + ex + f, the 5th differences would be the differences between the coefficients of the x^5 terms.
Now, the statement claims that the 5th differences of a 5th-degree polynomial are a constant value. Let's consider an example to analyze this claim:
Suppose we have a polynomial sequence: 2, 6, 12, 20, 30, 42. The differences would be: 6-2=4, 12-6=6, 20-12=8, 30-20=10, 42-30=12. And the second differences would be: 6-4=2, 8-6=2, 10-8=2, 12-10=2.
From this example, we can see that the second differences (which are the 5th differences for a 5th-degree polynomial) are indeed a constant value (in this case, 2).
Therefore, the statement is true. The 5th differences of a 5th-degree polynomial are a constant value.
So, the correct answer is a. True.