Answered

Get personalized answers to your specific questions with IDNLearn.com. Join our Q&A platform to access reliable and detailed answers from experts in various fields.

What is the solution to [tex]$x^3+4 x^2\ \textgreater \ x+4$[/tex]?

A. [tex]-4\ \textless \ x\ \textless \ -1[/tex]

B. [tex]-4\ \textless \ x\ \textless \ 1[/tex]

C. [tex]-4\ \textless \ x\ \textless \ -1[/tex] or [tex]x\ \textgreater \ 1[/tex]

D. [tex]x\ \textless \ -4[/tex] or [tex]-1\ \textless \ x\ \textless \ 1[/tex]

Sagot :

GinnyAnsweridn
To solve the inequality [tex]\( x^3 + 4x^2 > x + 4 \)[/tex], we follow these steps:

1. Move all terms to one side of the inequality:
Subtract [tex]\(x + 4\)[/tex] from both sides to get a single expression on one side:

[tex]\[ x^3 + 4x^2 - x - 4 > 0 \][/tex]

2. Factor the polynomial if possible:
The polynomial [tex]\(x^3 + 4x^2 - x - 4\)[/tex] is not easily factorable, so we need to use other methods such as finding roots through polynomial solving techniques or graphing.

3. Determine the critical points:
Find the roots of the equation [tex]\( x^3 + 4x^2 - x - 4 = 0 \)[/tex]. These roots are points where the polynomial changes sign. Solving for these roots:

Solving [tex]\( x^3 + 4x^2 - x - 4 = 0 \)[/tex] gives us:

[tex]\[ x = -4, x = -1, \text{ and } x = 1 \][/tex]

4. Plot these critical points on a number line:
These points split the number line into intervals. The intervals to test are:
- [tex]\( (-\infty, -4) \)[/tex]
- [tex]\( (-4, -1) \)[/tex]
- [tex]\( (-1, 1) \)[/tex]
- [tex]\( (1, \infty) \)[/tex]

5. Test each interval:
We pick test points from each interval to determine the sign of the polynomial in those intervals.

For interval [tex]\( (-\infty, -4) \)[/tex]:
- Choose [tex]\( x = -5 \)[/tex], [tex]\( x^3 + 4x^2 - x - 4 = -125 + 100 + 5 - 4 = -24 \)[/tex] (negative)

For interval [tex]\( (-4, -1) \)[/tex]:
- Choose [tex]\( x = -2 \)[/tex], [tex]\( x^3 + 4x^2 - x - 4 = -8 + 16 + 2 - 4 = 6 \)[/tex] (positive)

For interval [tex]\( (-1, 1) \)[/tex]:
- Choose [tex]\( x = 0 \)[/tex], [tex]\( x^3 + 4x^2 - x - 4 = 0 + 0 - 0 - 4 = -4 \)[/tex] (negative)

For interval [tex]\( (1, \infty) \)[/tex]:
- Choose [tex]\( x = 2 \)[/tex], [tex]\( x^3 + 4x^2 - x - 4 = 8 + 16 - 2 - 4 = 18 \)[/tex] (positive)

6. Combine the intervals where the polynomial is positive:
Based on our testing:
- [tex]\( x \in (-4, -1) \)[/tex]
- [tex]\( x \in (1, \infty) \)[/tex]

These are the intervals where [tex]\(x^3 + 4x^2 - x - 4 > 0\)[/tex].

Thus, the solution to the inequality [tex]\( x^3 + 4x^2 > x + 4 \)[/tex] is:

[tex]\[ -4 < x < -1 \quad \text{or} \quad x > 1 \][/tex]

Therefore, the correct answer is:

[tex]\[ -4 < x < -1 \quad \text{or} \quad x > 1 \][/tex]
We appreciate every question and answer you provide. Keep engaging and finding the best solutions. This community is the perfect place to learn and grow together. Your questions are important to us at IDNLearn.com. Thanks for stopping by, and come back for more reliable solutions.