WHAT is the answer of (x-4) power 4 ?

METHOD 1:

If you are familiar with the expansion of (a + b)^n then this is straightforward. Captain Matticus has detailed this approach.

Each term is C(n,k) * a^(n-k) * b^k

a = x

b = -4

Term 0:

C(4,0) * x^4 * (-4)^0 --> 1 * x^4 * 1 = x^4

Term 1:

C(4,1) * x^3 * (-4)^1 --> 4 * x^3 * -4 = -16x^3

Term 2:

C(4,2) * x^2 * (-4)^2 --> 6 * x^2 * 16 = 96x²

Term 3:

C(4,3) * x^1 * (-4)^3 --> 4 * x * 64 = 256x

Term 4:

C(4,4) * x^0 * (-4)^4 --> 1 * 1 * 256 = 256

Putting that together:

x^4 - 16x^3 + 96x² - 256x + 256

METHOD 2:

Barring that, you can also just multiply it out manually.

(x - 4)^4

= (x - 4)(x - 4)(x - 4)(x - 4)

Multiply a pair of binomials using distributing or FOIL --> (x - 4)(x - 4) = x² - 8x + 16

= (x² - 8x + 16)(x² - 8x + 16)

Then multiply these together by distributing:

x²(x² - 8x + 16) - 8x(x² - 8x + 16) + 16(x² - 8x + 16)

x^4 - 8x^3 + 16x² - 8x^3 + 64x² - 128x + 16x² - 128x + 256

Then group like terms:

x^4 + (-8x^3 - 8x^3) + (16x² + 64x² + 16x²) + (-128x + -128x) + 256

Answer:

x^4 - 16x^3 + 96x² - 256x + 256

x^4-16x^3+96x^2-256x+256

Just multiply out each term.

(a - b)^4

= a^4 - 4 a^3 b + 6 a^2 b^2 - 4 a b^3 + b^4

(x - 4)^4

= x^4 - 16 x^3 + 96 x^2 - 256 x + 256

(x^2-8x+16)^2 =

x^4 - 16x^3 + 96x² - 256x + 256

Answer

( x - 4 ) ² ( x - 4 ) ²

( x² - 8x + 16 ) ( x² - 8x + 16 )

x⁴ - 8x³ + 16x²

___- 8x³ + 64x² - 128 x

__________16x² - 128x + 256

x⁴ - 16x³ + 96x² - 256x + 256

(x - 4)^4 = x^4 - 4(4^1)(x^3) + 6(4^2)(x^2) - 4(4^3)(x) + 4^4

x^4 - 16x^3 +96x^2 - 265x + 256

(a + b)^n = sigma((n! / (k! * (n - k)!)) * a^(n - k) * b^n , k = 0 , k = n)

(x - 4)^4 =>

(4! / (0! * (4 - 0)!)) * x^(4 - 0) * (-4)^(0) + (4! / (1! * (4 - 1)!)) * x^(4 - 1) * (-4)^1 + (4! / (2! * (4 - 2)!)) * x^(4 - 2) * (-4)^2 + (4! / (3! * (4 - 3)!)) * x^(4 - 3) * (-4)^3 + (4! / (4! * (4 - 4)!)) * x^(4 - 4) * (-4)^4 =>

(24 / (1 * 24)) * x^(4) * 1 + (24 / (1 * 6)) * x^(3) * (-4) + (24 / (2 * 2)) * x^(2) * (-4)^2 + (24 / (6 * 1)) * x^(1) * (-64) + (24 / (24 * 1)) * x^(0) * (256) =>

1 * x^(4) + 4 * (-4) * x^(3) + 6 * (16) * x^(2) + 4 * (-64) * x + 1 * 1 * 256 =>

x^(4) - 16 * x^(3) + 96 * x^(2) - 256 * x + 256

x^4 - 16 x^3 + 96 x^2 - 256 x + 256