(2x+1)(4x^2-2x+1)-8(x+1/2)(x^2-1/2x+1/4)

4 min read Jun 16, 2024

Simplifying the Expression (2x+1)(4x^2-2x+1)-8(x+1/2)(x^2-1/2x+1/4)

This expression involves the product of two binomials and two trinomials. To simplify it, we'll use the following strategies:

Recognize patterns: Both trinomials within the expression are in the form of a^3 + b^3, where a = 2x and b = 1.
Apply the sum of cubes factorization: We know that a^3 + b^3 = (a + b)(a^2 - ab + b^2).
Simplify and combine terms: After applying the factorization, we'll combine like terms to arrive at the simplified expression.

Let's break down the steps:

Step 1: Recognize Patterns

(4x^2 - 2x + 1): This trinomial is in the form of (2x)^3 - (1)^3, which is a^3 + b^3.
(x^2 - 1/2x + 1/4): This trinomial is in the form of (x)^3 - (1/2)^3, which is also a^3 + b^3.

Step 2: Apply Sum of Cubes Factorization

(2x+1)(4x^2-2x+1): Applying the factorization, we get (2x + 1)((2x)^2 - (2x)(1) + (1)^2) = (2x + 1)(4x^2 - 2x + 1).
(x+1/2)(x^2-1/2x+1/4): Applying the factorization, we get (x + 1/2)((x)^2 - (x)(1/2) + (1/2)^2) = (x + 1/2)(x^2 - 1/2x + 1/4).

Step 3: Simplify and Combine Terms

Now, we have: (2x + 1)(4x^2 - 2x + 1) - 8(x + 1/2)(x^2 - 1/2x + 1/4)

Notice that the first term is already simplified, but we can simplify the second term by distributing the 8:

= (2x + 1)(4x^2 - 2x + 1) - (8x + 4)(x^2 - 1/2x + 1/4)

Now, we can expand both products:

= 8x^3 - 4x^2 + 2x + 4x^2 - 2x + 1 - (8x^3 - 4x^2 + 2x + 4x^2 - 2x + 1)

Combining like terms, we get:

= 8x^3 - 4x^2 + 2x + 4x^2 - 2x + 1 - 8x^3 + 4x^2 - 2x - 4x^2 + 2x - 1

= 0

Therefore, the simplified form of the expression (2x+1)(4x^2-2x+1)-8(x+1/2)(x^2-1/2x+1/4) is simply 0.