Solve each polynomial inequality in Exercises 1–42 and graph the solution set on a real number line. Express each solution set in interval notation. $x^{3} - 3 x^{2} - 9 x + 27 < 0$

검증된 단계별 안내

Start by rewriting the inequality: $x^3 - 3x^2 - 9x + 27 < 0$.

Factor the cubic polynomial if possible. Look for rational roots using the Rational Root Theorem and test possible roots such as $\pm1$, $\pm3$, $\pm9$, $\pm27$.

Once a root is found, use polynomial division or synthetic division to factor the cubic into a product of a linear factor and a quadratic factor.

Solve the quadratic factor by factoring further or using the quadratic formula to find all critical points (roots) where the expression equals zero.

Use the critical points to divide the real number line into intervals, then test a point from each interval in the original inequality to determine where the polynomial is less than zero. Express the solution set in interval notation.

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주요 개념

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Polynomial Inequalities

Polynomial inequalities involve expressions where a polynomial is compared to zero or another value using inequality signs. Solving them requires finding the values of the variable that make the inequality true, often by analyzing the sign of the polynomial over different intervals.

Factoring Polynomials

Factoring is the process of breaking down a polynomial into simpler polynomials (factors) that multiply to give the original. Factoring helps identify the roots or zeros of the polynomial, which are critical points for determining where the polynomial changes sign.

Interval Notation and Sign Analysis

Interval notation expresses solution sets as ranges of values on the number line. After finding the roots, sign analysis tests the polynomial's sign in intervals between roots to determine where the inequality holds, allowing the solution to be written accurately in interval notation.

Solve each polynomial inequality in Exercises 1–42 and graph the solution set on a real number line. Express each solution set in interval notation. x3−3x2−9x+27<0x^3−3x^2−9x+27<0