What is the sum of all coefficients of the polynomial?
f(x)= (3x^3 - 3x - 1)^38 * (2x+1)^4 * (x^4 - 3x^2 + 5x - 2)^77
To find the sum of all coefficients of a polynomial, you need to expand and simplify the polynomial expression.
In this case, we have three polynomial expressions multiplied together:
1. (3x^3 - 3x - 1)^38
2. (2x + 1)^4
3. (x^4 - 3x^2 + 5x - 2)^77
To expand these expressions, we need to apply the binomial theorem.
1. (3x^3 - 3x - 1)^38:
Applying the binomial theorem, we can expand this expression as follows:
(3x^3 - 3x - 1)^38 = Sum of ((38 choose k) * (3x^3)^k * (-3x)^{38-k} * (-1)^(38-k)), for k = 0 to 38.
To calculate the sum of coefficients, we only need the constant term, which is the term that doesn't have any x variables. In this case, the constant term is (-1)^38 = 1.
2. (2x + 1)^4:
We can expand this expression using the binomial theorem:
(2x + 1)^4 = Sum of ((4 choose k) * (2x)^k * (1)^(4-k)), for k = 0 to 4.
Again, we only need the constant term, which is (1)^(4 - 0) = 1.
3. (x^4 - 3x^2 + 5x - 2)^77:
Expanding this expression using the binomial theorem:
(x^4 - 3x^2 + 5x - 2)^77 = Sum of ((77 choose k) * (x^4)^k * (-3x^2)^{77-k} * (5x)^k * (-2)^{77-k}), for k = 0 to 77.
Once again, we are interested in the constant term, which is (-2)^(77 - 0) = 1.
To find the sum of all coefficients, we multiply the constant terms from each of the expanded expressions:
Sum of coefficients = 1 * 1 * 1 = 1.
Therefore, the sum of all coefficients of the given polynomial is 1.