Find the coefficient of x^4 in the expansion of (4x-1)^5

The coeff. of x^4 in this expansion is -1280.

Step-by-step explanation:

Start by writing down a Pascal's Triangle:

               1

           1      1

        1       2     1

     1       3     3      1

  1      4      6     4       1

1      5     10    10     5      1

Example:  write out (x + y)^3:

Use the 4th row of Pascal's Triangle:  1(x^3) + 3(x^2) + 3(x^1) + 1.  Note how

the powers of x decrease from 3 through 2, 1 and 0.

Now let's apply this to the problem at hand.   Use the coefficients in the 6th row of the Triangle, above:

1[4x]^5 + 5[4x]^4·(-1) +

The first term is 1[4x]^5, or [4x]^5, or 4^5·x^5, or 1024·x^5.

The second term is 5[4x]^4·(-1), or 5·4^4·(-1), or 5[256](-1) = -1280.

Thus, the coeff. of x^4 in this expansion is -1280.

a: 1/6, j: 1/4

step-by-step explanation: you are welcome

solution:

the given function is ,

      =

the three roots of this quadratic function is , 0 ,8 and 10.all the roots are real and rational.

if a quadratic equation is of the form ,

sum of the roots = a + b +c= 0+8+10=18=  

product of roots taken two at a time = a b + b c+ c a = 0+80+0=80 =  

product of roots = a b c = 0=  

there are 11 doors in a hall starting from 0 to 11.the entry in the hall is permitted through gate number 0 and exit is allowed through only gate number 8 and 10.so, three roots(0,8,10) of quadratic function is given by entry and exit route of this hall.