Question

Find the coefficient of x^4 and x^5 in the expansion of (4x - 2x^2)^6

Answer 1

Since we know:

$(a-b)^6 = a^6-6a^5b+15a^4b^2-20a^3b^3+15a^2b^4-6ab^5+b^6$

Applying it:

$(4x - 2x^2)^6$

$= (4x)^6-6(4x)^5(2x^2)+15(4x)^4(2x^2)^2-20(4x)^3(2x^2)^3+15(4x)^2(2x^2)^4-6(4x)(2x^2)^5+(2x^2)^6$

$=64x^{12}-768x^{11}+3840x^{10}-10240x^9+15360x^8 - 12288x^7+4096x^6$

Find the coefficient:

$\text{coefficient of } x^4 = 0$

$\text{coefficient of } x^5 = 0$

Answer 2

The formula for (a - b)^6 is

(a-b)^6 = a^6 - 6a^5b + 15a^4b^2- 20a^3b^3 + 15a^2b^4 - 6ab^5 + b^6

 (4x - 2x^2)^6

= (4x)^6 - 6(4x)^5(2x^2) + 15(4x)^4(2x^2)^2 - 20(4x)^3 (2x^2)^3 + 15(4x)^2 (2x^2)^4 - 6(4x)(2x^2)^5 + (2x^2)^6

64x^12 - 768x^11 + 3840x^10 - 10240x^9 + 15360x^8 - 12288x^7 + 4096x^6

There are no coefficients for x^4 and x^5