Answer :
To solve the inequality [tex]\(9x^3 + 45x^2 - 25x - 125 > 0\)[/tex], we can follow these steps:
1. Find the roots of the polynomial: First, we identify the roots of the equation [tex]\(9x^3 + 45x^2 - 25x - 125 = 0\)[/tex]. These roots will help divide the number line into intervals for testing the sign of the polynomial.
2. Factor the polynomial: It appears that the polynomial can be factored. To do this, we can use techniques like synthetic division or trial and error to find at least one rational root using the Rational Root Theorem.
3. Check potential roots: By testing simple integer roots (such as [tex]\(\pm 1, \pm 5, \pm \frac{125}{9}\)[/tex], etc.), particularly divisor factors of the last term and the leading coefficient, we can find at least one root.
4. Perform synthetic division: Once a root, say [tex]\(x = a\)[/tex], is found, factor the polynomial by dividing it by [tex]\(x - a\)[/tex]. Continue with the remaining quadratic part, which may need to be factored further using the quadratic formula, or may already be factorable.
5. Solve the factored polynomial: After fully factoring, set each factor greater than zero to find the solution intervals. This often comprises expressions like [tex]\((x - r_1), (x - r_2), (ax + b)\)[/tex], etc., where [tex]\(r_1, r_2\)[/tex] are roots.
6. Test intervals: The real line is divided into intervals using your found roots. Pick a test point from each interval and substitute it back into the inequality to check whether it satisfies the inequality.
Here's an outline of these steps for your specific problem:
1. Find rational roots: Test potential rational roots such as [tex]\(\pm 1, \pm 5\)[/tex], and others using the Rational Root Theorem.
2. Factor completely: If you find a root, divide to factor the polynomial completely.
3. Use synthetic division: Suppose [tex]\(x = -5\)[/tex] is a root; you can use synthetic division to factor the polynomial into simpler binomials or a quadratic expression.
4. Solve the inequality: After factoring, solve each part for when it is zero to find critical points, and test intervals to check if these intervals satisfy the inequality.
Once the intervals are determined, your final answer will be a union of all the intervals where the inequality holds true.
At each step, double-check calculations to confirm the factorization and polarity of the intervals. This ensures a good understanding of both technical and theoretical approaches are applied.
1. Find the roots of the polynomial: First, we identify the roots of the equation [tex]\(9x^3 + 45x^2 - 25x - 125 = 0\)[/tex]. These roots will help divide the number line into intervals for testing the sign of the polynomial.
2. Factor the polynomial: It appears that the polynomial can be factored. To do this, we can use techniques like synthetic division or trial and error to find at least one rational root using the Rational Root Theorem.
3. Check potential roots: By testing simple integer roots (such as [tex]\(\pm 1, \pm 5, \pm \frac{125}{9}\)[/tex], etc.), particularly divisor factors of the last term and the leading coefficient, we can find at least one root.
4. Perform synthetic division: Once a root, say [tex]\(x = a\)[/tex], is found, factor the polynomial by dividing it by [tex]\(x - a\)[/tex]. Continue with the remaining quadratic part, which may need to be factored further using the quadratic formula, or may already be factorable.
5. Solve the factored polynomial: After fully factoring, set each factor greater than zero to find the solution intervals. This often comprises expressions like [tex]\((x - r_1), (x - r_2), (ax + b)\)[/tex], etc., where [tex]\(r_1, r_2\)[/tex] are roots.
6. Test intervals: The real line is divided into intervals using your found roots. Pick a test point from each interval and substitute it back into the inequality to check whether it satisfies the inequality.
Here's an outline of these steps for your specific problem:
1. Find rational roots: Test potential rational roots such as [tex]\(\pm 1, \pm 5\)[/tex], and others using the Rational Root Theorem.
2. Factor completely: If you find a root, divide to factor the polynomial completely.
3. Use synthetic division: Suppose [tex]\(x = -5\)[/tex] is a root; you can use synthetic division to factor the polynomial into simpler binomials or a quadratic expression.
4. Solve the inequality: After factoring, solve each part for when it is zero to find critical points, and test intervals to check if these intervals satisfy the inequality.
Once the intervals are determined, your final answer will be a union of all the intervals where the inequality holds true.
At each step, double-check calculations to confirm the factorization and polarity of the intervals. This ensures a good understanding of both technical and theoretical approaches are applied.
