for the expansion of (k+t)power22, state
a. the number of terms
b. the degree of each term
c. the first four terms in the expansion, without coeffiecients
This is a question of introduction to the Binomial Theorem.
If you are not in a classroom or have no textbook, here is a short summary to fit your problem
http://www.intmath.com/Series-binomial-theorem/4_Binomial-theorem.php
To find the expansion of the expression (k + t)^22, we can use the Binomial Theorem. The Binomial Theorem states that for any positive integer n:
(a + b)^n = C(n, 0)a^n b^0 + C(n, 1)a^n-1 b^1 + C(n, 2)a^n-2 b^2 + ... + C(n, n-1)a^1 b^n-1 + C(n, n)a^0 b^n
where C(n, r) represents the binomial coefficient, which is calculated as C(n, r) = n! / (r!(n - r)!), and "!" denotes factorial.
Now, let's apply the Binomial Theorem to find the expansion of (k + t)^22.
a. The number of terms:
According to the Binomial Theorem, the expansion of (k + t)^22 will have 23 terms. The number of terms is always equal to (n + 1), where n is the power to which the binomial is raised.
b. The degree of each term:
The degree of each term in the expansion is equal to the sum of the exponents of the variables. In this case, the expanded form will have terms with k raised to decreasing powers from 22 down to 0, while the power of t will increase from 0 to 22. Therefore, the degree of each term will range from 0 to 22.
c. The first four terms in the expansion, without coefficients:
The expanded form without coefficients will involve only the variables k and t raised to their respective powers. Let's write down the first four terms of the expansion:
1. Term 1: (k^22)(t^0)
2. Term 2: (k^21)(t^1)
3. Term 3: (k^20)(t^2)
4. Term 4: (k^19)(t^3)
So, the first four terms in the expansion, without coefficients, are:
1. k^22
2. k^21 * t
3. k^20 * t^2
4. k^19 * t^3
Remember, these terms only represent the variable parts of the expansion without the coefficients.