\( (x+2 y)^{3}= \) 25. \( (2 a+b)^{3}= \) 27. \( (2 x-5)^{3}= \)

To expand the expressions using the binomial theorem, we can use the formula for the cube of a binomial: \[ (a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3 \] Now, let's apply this to each of the given expressions. 1. **For \( (x + 2y)^3 \)**: Here, \( a = x \) and \( b = 2y \). \[ (x + 2y)^3 = x^3 + 3(x^2)(2y) + 3(x)(2y)^2 + (2y)^3 \] Now, calculate each term: - \( x^3 \) - \( 3(x^2)(2y) = 6x^2y \) - \( 3(x)(2y)^2 = 3(x)(4y^2) = 12xy^2 \) - \( (2y)^3 = 8y^3 \) Putting it all together: \[ (x + 2y)^3 = x^3 + 6x^2y + 12xy^2 + 8y^3 \] 2. **For \( (2a + b)^3 \)**: Here, \( a = 2a \) and \( b = b \). \[ (2a + b)^3 = (2a)^3 + 3(2a)^2b + 3(2a)b^2 + b^3 \] Now, calculate each term: - \( (2a)^3 = 8a^3 \) - \( 3(2a)^2b = 3(4a^2)b = 12a^2b \) - \( 3(2a)b^2 = 6ab^2 \) - \( b^3 = b^3 \) Putting it all together: \[ (2a + b)^3 = 8a^3 + 12a^2b + 6ab^2 + b^3 \] 3. **For \( (2x - 5)^3 \)**: Here, \( a = 2x \) and \( b = -5 \). \[ (2x - 5)^3 = (2x)^3 + 3(2x)^2(-5) + 3(2x)(-5)^2 + (-5)^3 \] Now, calculate each term: - \( (2x)^3 = 8x^3 \) - \( 3(2x)^2(-5) = 3(4x^2)(-5) = -60x^2 \) - \( 3(2x)(-5)^2 = 3(2x)(25) = 150x \) - \( (-5)^3 = -125 \) Putting it all together: \[ (2x - 5)^3 = 8x^3 - 60x^2 + 150x - 125 \] In summary, the expanded forms are: 1. \( (x + 2y)^3 = x^3 + 6x^2y + 12xy^2 + 8y^3 \) 2. \( (2a + b)^3 = 8a^3 + 12a^2b + 6ab^2 + b^3 \) 3. \( (2x - 5)^3 = 8x^3 - 60x^2 + 150x - 125 \)