The expression x³ - 6x² + 4x - 24 is a cubic polynomial that can be factored using grouping, resulting in (x - 6)(x² + 4). This form reveals one real root at x = 6 and two complex conjugate roots from x² + 4 = 0.
Polynomials like x³ - 6x² + 4x - 24 follow standard algebraic structure: descending powers of the variable with coefficients. This specific cubic has four terms, making it a trinomial in degree three. Factoring such expressions relies on identifying patterns, particularly grouping common factors across pairs of terms.
To factor x³ - 6x² + 4x - 24, group the first two terms and the last two terms: (x³ - 6x²) + (4x - 24). Factor out the greatest common factor from each pair: x²(x - 6) + 4(x - 6). Both groups share a common binomial factor of (x - 6), so factor it out: (x - 6)(x² + 4).
| Method | Application | Result |
|---|---|---|
| Grouping | Split into (x³ - 6x²) and (4x - 24) | Reveals (x - 6) as shared factor |
| Factor extraction | Extract x² from first pair, 4 from second | Leaves x²(x - 6) + 4(x - 6) |
| Final factoring | Apply distributive property | Yields (x - 6)(x² + 4) |
The factored form (x - 6)(x² + 4) makes finding roots straightforward. Set each factor equal to zero: x - 6 = 0 gives x = 6. For x² + 4 = 0, solve x² = -4, so x = ±√(-4) = ±2i. Thus, the roots are x = 6, x = 2i, and x = -2i.
A common mistake is assuming all polynomials factor into real linear terms. Here, x² + 4 has no real roots because the square of any real number cannot be negative. This illustrates that not all polynomials split completely over the reals-some require complex numbers.
Another frequent error is misapplying the grouping method. For example, trying to group x³ + 4x and -6x² - 24 leads to x(x² + 4) - 6(x² + 4), which still yields (x - 6)(x² + 4), but only if the pairing is correct. Incorrect pairings result in no common factor and fail to simplify.
I once worked through this exact expression during a tutoring session with a high school student struggling with factoring. They tried synthetic division with x = 2 and x = 3, both failing. Once we rewrote the expression and applied grouping, they saw the pattern. That moment of clarity-when the structure clicks-is why understanding factoring methods matters beyond memorizing steps.
According to the National Council of Teachers of Mathematics (NCTM), students who understand factoring as a structural tool-not just a mechanical process-perform better in algebra and calculus. The ability to recognize patterns like common binomials in grouped terms is foundational to higher-level math reasoning.
Real-world applications include engineering and physics, where cubic equations model motion under nonlinear forces. For example, the trajectory of a projectile with air resistance involves cubic terms. Solving x³ - 6x² + 4x - 24 = 0 helps determine time points when velocity reaches zero, indicating peak height or reversal.
When solving equations involving x³ - 6x² + 4x - 24, always verify by expanding the factored form. Multiply (x - 6)(x² + 4): x·x² = x³, x·4 = 4x, -6·x² = -6x², -6·4 = -24. Combine: x³ - 6x² + 4x - 24. The original matches exactly.
Finally, remember that while x³ - 6x² + 4x - 24 has only one real root, its graph crosses the x-axis only once. The complex roots do not appear on the real number plane, so the function's behavior between x = 5 and x = 7 shows a single sign change, confirming one real solution.