Solve each cubic equation using factoring and the quadratic formula. See Example 7.
$x^{3} - 27 = 0$

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Recognize that the equation $x^3 - 27 = 0$ is a difference of cubes, since $27$ can be written as \$3^3$.

Recall the difference of cubes factoring formula: $a^3 - b^3 = (a - b)(a^2 + ab + b^2)$.

Apply the formula with $a = x$ and $b = 3$ to factor the equation as $(x - 3)(x^2 + 3x + 9) = 0$.

Set each factor equal to zero: $x - 3 = 0$ and $x^2 + 3x + 9 = 0$.

Solve the linear equation $x - 3 = 0$ for $x$, and then solve the quadratic equation $x^2 + 3x + 9 = 0$ using the quadratic formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ where $a=1$, $b=3$, and $c=9$.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Factoring Cubic Equations

Factoring cubic equations involves expressing the cubic polynomial as a product of simpler polynomials, often starting with recognizing special forms like the difference of cubes. For example, x³ - 27 can be factored using the formula a³ - b³ = (a - b)(a² + ab + b²). This step simplifies solving the equation by breaking it into lower-degree factors.

Difference of Cubes Formula

The difference of cubes formula states that a³ - b³ = (a - b)(a² + ab + b²). This identity helps factor expressions like x³ - 27 by identifying a = x and b = 3. Applying this formula transforms the cubic into a linear factor and a quadratic factor, making it easier to find all roots of the equation.

Quadratic Formula

The quadratic formula, x = [-b ± √(b² - 4ac)] / (2a), is used to solve quadratic equations that cannot be factored easily. After factoring the cubic into a linear and quadratic factor, the quadratic formula helps find the roots of the quadratic part, ensuring all solutions to the original cubic equation are found.

Solve each cubic equation using factoring and the quadratic formula. See Example 7.x3−27=0x^3 - 27 = 0