(2x+3)^3(4x-1)^2
To expand the expression (2x+3)^3(4x-1)^2, you need to apply the binomial theorem. The binomial theorem states that for any two terms a and b raised to powers n and m respectively, the expanded form is given by:
(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-2)a^2 * b^(n-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, which is calculated as n! / (k!(n-k)!), and n! denotes the factorial of n.
Let's apply this theorem to expand (2x+3)^3 first:
(2x+3)^3 = C(3, 0)(2x)^3 * 3^0 + C(3, 1)(2x)^(3-1) * 3^1 + C(3, 2)(2x)^(3-2) * 3^2 + C(3, 3)(2x)^0 * 3^3
Simplifying this expression:
C(3, 0)(2x)^3 * 3^0 = (1)(8x^3)(1) = 8x^3
C(3, 1)(2x)^(3-1) * 3^1 = (3)(4x^2)(3) = 36x^2
C(3, 2)(2x)^(3-2) * 3^2 = (3)(2x)(9) = 54x
C(3, 3)(2x)^0 * 3^3 = (1)(1)(27) = 27
Therefore, (2x+3)^3 expands to 8x^3 + 36x^2 + 54x + 27.
Now, let's expand (4x-1)^2:
(4x-1)^2 = C(2, 0)(4x)^2 * (-1)^0 + C(2, 1)(4x)^(2-1) * (-1)^1 + C(2, 2)(4x)^0 * (-1)^2
Simplifying this expression:
C(2, 0)(4x)^2 * (-1)^0 = (1)(16x^2)(1) = 16x^2
C(2, 1)(4x)^(2-1) * (-1)^1 = (2)(4x)(-1) = -8x
C(2, 2)(4x)^0 * (-1)^2 = (1)(1)(1) = 1
Therefore, (4x-1)^2 expands to 16x^2 - 8x + 1.
Finally, multiply the two expressions we obtained:
(2x+3)^3(4x-1)^2 = (8x^3 + 36x^2 + 54x + 27)(16x^2 - 8x + 1)
To simplify the expression further, you can perform multiplication of each term using the distributive property:
(8x^3 + 36x^2 + 54x + 27)(16x^2 - 8x + 1) = 8x^3(16x^2 - 8x + 1) + 36x^2(16x^2 - 8x + 1) + 54x(16x^2 - 8x + 1) + 27(16x^2 - 8x + 1)
Now, distribute and simplify each term:
8x^3(16x^2 - 8x + 1) = 128x^5 - 64x^4 + 8x^3
36x^2(16x^2 - 8x + 1) = 576x^4 - 288x^3 + 36x^2
54x(16x^2 - 8x + 1) = 864x^3 - 432x^2 + 54x
27(16x^2 - 8x + 1) = 432x^2 - 216x + 27
Combining all the terms, we have:
(2x+3)^3(4x-1)^2 = 128x^5 - 64x^4 + 8x^3 + 576x^4 - 288x^3 + 36x^2 + 864x^3 - 432x^2 + 54x + 432x^2 - 216x + 27
Simplifying further:
128x^5 + 576x^4 - 64x^4 - 288x^3 + 8x^3 + 864x^3 + 36x^2 - 432x^2 + 432x^2 + 54x - 216x + 27
Combine like terms:
128x^5 + 512x^4 + 584x^3 + 36x^2 - 162x + 27
Therefore, the expanded form of (2x+3)^3(4x-1)^2 is 128x^5 + 512x^4 + 584x^3 + 36x^2 - 162x + 27.