Question

Expand (3x1 + 2x2 + 5x3)^4

Answer 1

To expand (3x1 + 2x2 + 5x3)^4, we can use the binomial theorem. The expanded form is 81x^4 + 108(2x2 + 5x3) + 54(2x2 + 5x3)^2 + 96x4 + 480x3 + 16x8 + 160x7 + 1600x6.

Step-by-step explanation:

To expand the expression (3x1 + 2x2 + 5x3)^4, we will use the binomial theorem. The binomial theorem states that (a + b)^n = a^n + nan-1b + (n(n-1))/2! a^n-2b^2 + ... + b^n. In this case, a = 3x1, b = 2x2 + 5x3, and n = 4. So we have (3x1 + 2x2 + 5x3)^4 = (3x1)^4 + 4(3x1)^3(2x2 + 5x3) + 6(3x1)^2(2x2 + 5x3)^2 + 4(3x1)(2x2 + 5x3)^3 + (2x2 + 5x3)^4.

Now we can simplify each term:

1. (3x1)^4 = 81x4 = 81x^4
2. 4(3x1)^3(2x2 + 5x3) = 4(27)(2x2 + 5x3) = 108(2x2 + 5x3)
3. 6(3x1)^2(2x2 + 5x3)^2 = 6(9)(2x2 + 5x3)^2 = 54(2x2 + 5x3)^2
4. 4(3x1)(2x2 + 5x3)^3 = 4(3x1)(8x4 + 40x3) = 4(24x4 + 120x3) = 96x4 + 480x3
5. (2x2 + 5x3)^4 = (4x4 + 40x3)(4x4 + 40x3) = 16x8 + 160x7 + 1600x6

Combining all these terms, we get the expanded form of (3x1 + 2x2 + 5x3)^4 as 81x^4 + 108(2x2 + 5x3) + 54(2x2 + 5x3)^2 + 96x4 + 480x3 + 16x8 + 160x7 + 1600x6.