To find the term independent of x in the expression [4x^3-2x^2]^5, we need to expand the expression using the binomial theorem and identify the term with x exponent equal to zero.
Using the binomial theorem, the expression [a+b]^n can be expanded as:
[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,k) represents the binomial coefficient "n choose k".
In our case, a = 4x^3 and b = -2x^2, and n = 5. Plugging these values into the formula, we get:
[4x^3-2x^2]^5 = C(5,0)*(4x^3)^5*(-2x^2)^0 + C(5,1)*(4x^3)^4*(-2x^2)^1 + C(5,2)*(4x^3)^3*(-2x^2)^2 + C(5,3)*(4x^3)^2*(-2x^2)^3 + C(5,4)*(4x^3)^1*(-2x^2)^4 + C(5,5)*(4x^3)^0*(-2x^2)^5.
Simplifying each term, we have:
[4x^3-2x^2]^5 = 1*(4x^3)^5*1 + 5*(4x^3)^4*(-2x^2)^1 + 10*(4x^3)^3*(-2x^2)^2 + 10*(4x^3)^2*(-2x^2)^3 + 5*(4x^3)^1*(-2x^2)^4 + 1*(4x^3)^0*(-2x^2)^5.
Now we can evaluate each term:
[4x^3-2x^2]^5 = (4x^3)^5 + 5*(4x^3)^4*(-2x^2) + 10*(4x^3)^3*(-2x^2)^2 + 10*(4x^3)^2*(-2x^2)^3 + 5*(4x^3)^1*(-2x^2)^4 + (-2x^2)^5.
Simplifying further, we get:
[4x^3-2x^2]^5 = 1024x^15 - 640x^10 + 240x^5 - 48x^10 + 32x^10 - 32x^10.
Combining like terms, we have:
[4x^3-2x^2]^5 = 1024x^15 - 656x^10 + 240x^5.
Therefore, the term independent of x is 240x^5.
Find the term independent of x[4x^3-2\x^2]^5
1 answer