(x+3)(x^2-3x+9)-(x-2)(x^2+2x+4)

Jasonbradley

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Jun 16, 2024

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Simplifying the Expression (x+3)(x^2-3x+9)-(x-2)(x^2+2x+4)

This expression involves the multiplication of two sets of binomials. The key to simplifying this expression lies in recognizing that both sets of binomials are in the form of (a+b)(a^2-ab+b^2) and (a-b)(a^2+ab+b^2) respectively, which are special cases of the difference of cubes formula.

Understanding the Difference of Cubes Formula:

The difference of cubes formula states: a³ - b³ = (a - b)(a² + ab + b²)

Similarly, the sum of cubes formula is: a³ + b³ = (a + b)(a² - ab + b²)

Applying the Formula to our Expression:

1. Identify a and b:

   • In the first set of binomials, a = x and b = 3.
   • In the second set of binomials, a = x and b = 2.

2. Apply the difference of cubes formula:

   • (x+3)(x^2-3x+9) = x³ + 3³ = x³ + 27
   • (x-2)(x^2+2x+4) = x³ - 2³ = x³ - 8

3. Substitute the simplified terms back into the original expression:

   • (x+3)(x^2-3x+9) - (x-2)(x^2+2x+4) = (x³ + 27) - (x³ - 8)

4. Simplify the expression:

   • x³ + 27 - x³ + 8 = 35

Therefore, the simplified expression (x+3)(x^2-3x+9)-(x-2)(x^2+2x+4) is equal to 35.