A sequence has constant 3rd differences of 1. How many terms are in its standard form polynomial expression?
The 3rd difference of a polynomial sequence is constant and equal to 1. This means that the degree of the polynomial must be 3.
In general, a polynomial of degree $n$ can be expressed as $P(x)=a_nx^n+a_{n-1}x^{n-1}+\dots+a_1x+a_0$, where $n$ is a nonnegative integer.
Therefore, a polynomial of degree 3 would be expressed as $P(x)=a_3x^3+a_2x^2+a_1x+a_0$.
Since there are 4 different terms in the polynomial ($a_3x^3$, $a_2x^2$, $a_1x$, and $a_0$), the standard form polynomial expression has $\boxed{4}$ terms.
The constant 3rd differences of 1 indicate that the sequence is generated by a cubic polynomial.
A cubic polynomial is defined by the standard form expression:
P(n) = an^3 + bn^2 + cn + d
Since the constant 3rd differences are 1, this means that the coefficient of the 3rd degree term (an^3) is 1.
Thus, the number of terms in the standard form polynomial expression is 4.
To determine the number of terms in the standard form polynomial expression of a sequence with constant third differences, we need to follow a step-by-step process. Here's how you can do it:
Step 1: Find the first differences of the sequence.
- Subtract each term from the next term in the sequence.
Step 2: Find the second differences.
- Subtract each first difference from the next first difference.
Step 3: Find the third differences.
- Subtract each second difference from the next second difference.
Step 4: Analyze the third differences.
- If the third differences are constant, like in this case (constant 3rd differences of 1), we can infer that the sequence can be represented by a polynomial expression.
Step 5: Determine the degree of the polynomial.
- Count the number of non-zero third differences to determine the degree of the polynomial.
Each non-zero third difference represents one term in the polynomial expression. So, the number of terms in the standard form polynomial expression is equal to the degree of the polynomial plus one.
In this case, since we have constant 3rd differences of 1, the degree of the polynomial is 3. Therefore, the number of terms in the standard form polynomial expression is:
3 (degree) + 1 = 4
So, the sequence can be represented by a 4-term standard form polynomial expression.