Problem 47 Solve. $$2 x^{3}+6 \leq 5 x^{2... [FREE SOLUTION]

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Precalculus: Graphs and Models, A Right Triangle Approach

Marvin L. Bittinger, Judith A. Beecher, David J. Ellenbogen

$Math Studyset 91影视 Explanations$ Math

6 Edition

Chapter 4: Problem 47

Solve. $$2 x^{3}+6 \leq 5 x^{2}+x$$

Short Answer

Expert verified

First, find the roots of $2x^3 - 5x^2 - x + 6 = 0$, and then test intervals around these roots to determine where the polynomial is non-positive.

Step by step solution

Move All Terms to One Side

Rewrite the inequality by moving all terms to one side of the equation so it becomes: \[ 2x^{3} - 5x^{2} - x + 6 \leq 0 \]

Find Critical Points

Solve the equation $2x^{3} - 5x^{2} - x + 6 = 0$ by finding its roots. You can use methods such as the Rational Root Theorem, synthetic division, or numerical methods to find the real roots.

Test Intervals

After identifying the roots, determine where the polynomial is less than or equal to zero by testing values from each interval created by the roots.

Write the Solution

Combine the intervals where the polynomial is less than or equal to zero to form the solution to the inequality.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Critical Points

A critical point in the context of polynomial inequalities is where the polynomial equals zero. These points divide the number line into separate intervals that we can test.

Critical points are important because they help us know where the polynomial might change its sign - going from positive to negative, or vice versa.

For example, in the equation:
\[ 2x^{3} - 5x^{2} - x + 6 = 0 \]
the critical points are the values of x that make the equation true.

Testing Intervals

Once you have the critical points, the next step is to test the intervals they create on the number line. Testing intervals help us determine where the polynomial expression is positive or negative.

Here鈥檚 a simple way to do it:

First, break the number line into intervals using the critical points.
Select a test point from each interval.
Substitute this test point into the polynomial.
Check the sign of the result (positive or negative).

If the inequality is \[2x^{3} - 5x^{2} - x + 6 \leq 0 \], we are interested in the intervals where the polynomial is less than or equal to zero.

This process will help us identify which intervals are part of the solution.

Solving Cubic Equations

Solving cubic equations, like the one in the problem, involves finding the roots of the equation. Cubic equations can be solved using various methods. Here are some common ones:

**Rational Root Theorem:** This theorem provides possible rational roots that can be tested.
**Synthetic Division:** Use this method to divide the polynomial by a candidate root. If it results in a zero remainder, the candidate is a root.
**Numerical Methods:** For more complex equations, numerical methods (like Newton's method) may be used to approximate roots.

For example, finding the roots of \[ 2x^{3} - 5x^{2} - x + 6 = 0 \], we would use one or more of these methods.

These roots or critical points then allow us to find the intervals for solving the original inequality.

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91影视

Solve. $$2 x^{3}+6 \leq 5 x^{2}+x$$

Short Answer

Step by step solution

Move All Terms to One Side

Find Critical Points

Test Intervals

Write the Solution

Key Concepts

Critical Points

Testing Intervals

Solving Cubic Equations

One App. One Place for Learning.

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