Problem 62 - Algebra and Trigonometry Real Mathematics, Real People

Question

Determine the intervals on which the polynomial is entirely negative and those on which it is entirely positive. $$3 x^{2}+8 x+6$$

Step-by-Step Solution

Answer

The polynomial \(3x^2 + 8x + 6\) is positive when \(x<-2\) or \(x>-1\), and negative in the interval \(-2

Step 1: Setting the polynomial equal to zero

To find the roots of the polynomial, set the polynomial \(3x^2+8x+6\) equal to zero and solve for \(x\).

Step 2: Solving the equation

The roots of the polynomial are the solutions of the equation \(3x^2+8x+6 = 0\). Use the quadratic formula \(-b \pm \sqrt{b^2-4ac}\over2a\), where \(a = 3\), \(b = 8\), and \(c = 6\). Substituting these values in the equation yields the roots \(x = -1 \) and \(x = -2\).

Step 3: Determine the intervals

Now that the roots are known we can use them to determine the intervals requested. We'll probe the intervals \(-\infty,-2\), \(-2,-1\), and \(-1,\infty\). To do this we can use any number within these intervals and substitute it in our polynomial equation. If the result is negative then the polynomial is negative in that interval. The interval is positive if the result is positive.

Step 4: Interval testing

Using a convenient number from each interval (-3 in \(-\infty,-2\), -1.5 in \(-2,-1\), and 0 in \(-1,\infty\)) we see that these yield positive, negative and positive results respectively. Therefore the polynomial is positive when \(x<-2\) or \(x>-1\), and it's negative when \(-2

Key Concepts

Quadratic Formula

The quadratic formula is a powerful tool used to find the roots of a quadratic equation. A quadratic equation is generally written in the form\[ ax^2 + bx + c = 0 \]where \(a\), \(b\), and \(c\) are constants. The quadratic formula is given by\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]This formula provides a straightforward method for solving any quadratic equation.
To use it, substitute the coefficients \(a\), \(b\), and \(c\) from your equation into the formula.
The expression under the square root, \(b^2 - 4ac\), is known as the discriminant. It determines the nature of the roots:

• If it is positive, the equation has two distinct real roots.
• If it is zero, there is one repeated real root.
• If it is negative, the roots are complex and not real.

For the polynomial \(3x^2 + 8x + 6\), plugging the values into the formula gives us roots at \(x = -1\) and \(x = -2\). These roots help in identifying polynomial behavior across intervals.

Polynomial Roots

Roots of a polynomial are the solutions to the equation where the polynomial equals zero.
In other words, they represent the values of \(x\) that make the polynomial\[ f(x) = 3x^2 + 8x + 6 \]equal to zero.
Finding the roots is essential as they act as the critical points that divide the polynomial into distinct intervals.
For our quadratic equation, the roots are found using the quadratic formula, resulting in \(x = -1\) and \(x = -2\).
Understanding roots allows us to assess the behavior of the polynomial across different x-values, which is crucial for identifying positive and negative intervals.
Once identified, these roots help us to categorize the polynomial's behavior on specific ranges separated by these points.

Interval Testing

Interval testing is a method used to determine the sign of a polynomial within intervals separated by its roots.
After calculating the roots of the polynomial, they divide the number line into segments, known as intervals.
For the polynomial \(3x^2 + 8x + 6\), the roots \(x = -1\) and \(x = -2\) create the following intervals:

• \((-\infty, -2)\)
• \((-2, -1)\)
• \((-1, \infty)\)

To determine on which of these intervals the polynomial is positive or negative, select a test point from each interval and substitute it into the original polynomial.
This process shows the behavior of the polynomial within those intervals, allowing us to determine where it is entirely positive and where it is negative.

Positive and Negative Intervals

Positive and negative intervals are sections on the number line where the polynomial takes either a positive or negative value respectively.
After determining the roots and testing intervals through substitution, it's revealed where the polynomial is positive or negative.
For the polynomial \(3x^2 + 8x + 6\):

• The interval \((-\infty, -2)\) yields a positive value when tested, indicating the polynomial is positive.
• The interval \((-2, -1)\) yields a negative value, showing the polynomial is negative there.
• The interval \((-1, \infty)\) again yields a positive result.

Therefore, the polynomial is positive when \(x < -2\) or \(x > -1\). It is negative between \(-2\) and \(-1\).
Identifying these intervals is key in applications like curve sketching, optimizing functions, and solving inequalities involving polynomials.

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Key Concepts

Quadratic Formula

The quadratic formula is a powerful tool used to find the roots of a quadratic equation. A quadratic equation is generally written in the form\[ ax^2 + bx + c = 0 \]where \(a\), \(b\), and \(c\) are constants. The quadratic formula is given by\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]This formula provides a straightforward method for solving any quadratic equation.
To use it, substitute the coefficients \(a\), \(b\), and \(c\) from your equation into the formula.
The expression under the square root, \(b^2 - 4ac\), is known as the discriminant. It determines the nature of the roots:

• If it is positive, the equation has two distinct real roots.
• If it is zero, there is one repeated real root.
• If it is negative, the roots are complex and not real.

For the polynomial \(3x^2 + 8x + 6\), plugging the values into the formula gives us roots at \(x = -1\) and \(x = -2\). These roots help in identifying polynomial behavior across intervals.

Polynomial Roots

Roots of a polynomial are the solutions to the equation where the polynomial equals zero.
In other words, they represent the values of \(x\) that make the polynomial\[ f(x) = 3x^2 + 8x + 6 \]equal to zero.
Finding the roots is essential as they act as the critical points that divide the polynomial into distinct intervals.
For our quadratic equation, the roots are found using the quadratic formula, resulting in \(x = -1\) and \(x = -2\).
Understanding roots allows us to assess the behavior of the polynomial across different x-values, which is crucial for identifying positive and negative intervals.
Once identified, these roots help us to categorize the polynomial's behavior on specific ranges separated by these points.

Interval Testing

Interval testing is a method used to determine the sign of a polynomial within intervals separated by its roots.
After calculating the roots of the polynomial, they divide the number line into segments, known as intervals.
For the polynomial \(3x^2 + 8x + 6\), the roots \(x = -1\) and \(x = -2\) create the following intervals:

• \((-\infty, -2)\)
• \((-2, -1)\)
• \((-1, \infty)\)

To determine on which of these intervals the polynomial is positive or negative, select a test point from each interval and substitute it into the original polynomial.
This process shows the behavior of the polynomial within those intervals, allowing us to determine where it is entirely positive and where it is negative.

Positive and Negative Intervals

Positive and negative intervals are sections on the number line where the polynomial takes either a positive or negative value respectively.
After determining the roots and testing intervals through substitution, it's revealed where the polynomial is positive or negative.
For the polynomial \(3x^2 + 8x + 6\):

• The interval \((-\infty, -2)\) yields a positive value when tested, indicating the polynomial is positive.
• The interval \((-2, -1)\) yields a negative value, showing the polynomial is negative there.
• The interval \((-1, \infty)\) again yields a positive result.

Therefore, the polynomial is positive when \(x < -2\) or \(x > -1\). It is negative between \(-2\) and \(-1\).
Identifying these intervals is key in applications like curve sketching, optimizing functions, and solving inequalities involving polynomials.