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

2 min read Jun 17, 2024
(x-1/2)(x+1/2)(4x-1)

Factoring and Expanding (x - 1/2)(x + 1/2)(4x - 1)

This expression involves factoring and expanding a polynomial. Here's a breakdown of the process:

Factoring

The expression is already factored, but we can simplify it further:

  • Recognize the Difference of Squares: The first two factors (x - 1/2) and (x + 1/2) are in the form of the difference of squares: (a - b)(a + b) = a² - b²
  • Apply the Formula: Applying this to our expression: (x - 1/2)(x + 1/2) = x² - (1/2)² = x² - 1/4
  • Combine with the Remaining Factor: Now our expression becomes: (x² - 1/4)(4x - 1)

Expanding

To expand the expression, we use the distributive property:

  • Multiply each term in the first factor by each term in the second factor:

    • x² * (4x - 1) = 4x³ - x²
    • -1/4 * (4x - 1) = -x + 1/4

  • Combine the terms: 4x³ - x² - x + 1/4

Final Result

Therefore, the expanded form of (x - 1/2)(x + 1/2)(4x - 1) is 4x³ - x² - x + 1/4.

Additional Notes

  • The expression can also be factored completely by factoring out a common factor of 1/4 from the expanded form.
  • This expression represents a cubic polynomial. Cubic polynomials have a degree of 3, meaning the highest power of the variable is 3.

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