If (2x-1)^4 = ax^4+bx^3+cx^2+dx+e then find the value of a+2b +4c+8d+16e..plz help easy way..
To find the value of a+2b+4c+8d+16e, we need to expand the expression (2x-1)^4 and then use the coefficients of the x terms.
Expanding (2x-1)^4 can be done using the binomial theorem, but there is an easier way if we observe the pattern:
(2x-1)^4 = (2x-1)(2x-1)(2x-1)(2x-1)
We can see that each term of the expansion will be of the form 2^a * (-1)^b * x^c, where a+b+c = 4.
Let's break it down term by term:
- The first term will have a = 4, b = 0, and c = 0. So the coefficient of this term is 2^4 * (-1)^0 = 16.
- The second term will have a = 3, b = 1, and c = 0. So the coefficient of this term is 2^3 * (-1)^1 = -8.
- The third term will have a = 2, b = 2, and c = 0. So the coefficient of this term is 2^2 * (-1)^2 = 4.
- The fourth term will have a = 2, b = 1, and c = 1. So the coefficient of this term is 2^2 * (-1)^1 = -4.
- The fifth term will have a = 1, b = 3, and c = 0. So the coefficient of this term is 2^1 * (-1)^3 = -2.
- The sixth term will have a = 1, b = 2, and c = 1. So the coefficient of this term is 2^1 * (-1)^2 = 2.
- The seventh term will have a = 1, b = 1, and c = 2. So the coefficient of this term is 2^1 * (-1)^1 = -2.
- The eighth term will have a = 1, b = 0, and c = 3. So the coefficient of this term is 2^1 * (-1)^0 = 2.
- The ninth term will have a = 0, b = 4, and c = 0. So the coefficient of this term is 2^0 * (-1)^4 = 1.
Now, we can see that the desired value a+2b+4c+8d+16e is: 16 - 16 + 4 - 4 - 2 + 2 - 2 + 2 + 16 = 16.
Therefore, the value of a+2b+4c+8d+16e is 16.
just use the binomial theorem
(2x-1)^4 = (2x)^4 - 4(2x)^3 + 6(2x)^2 - 4(2x) + 1
Now just match up the coefficients with a,b,c,d,e and evaluate your expression