Problem 54 - Elementary Algebra

Question

$$ m^{3}+8 m^{2}+15 m=0 $$

Step-by-Step Solution

Answer

The solutions are: $$m = 0, -3, -5.$$

Step 1 - Identify the Polynomial

Recognize the given equation is a polynomial in the form of a cubic function: $$m^{3} + 8m^{2} + 15m = 0.$$

Step 2 - Factor Out the Greatest Common Factor (GCF)

First, factor out the common term, which is 'm' in this case: $$m (m^{2} + 8 m + 15) = 0.$$

Step 3 - Factor the Quadratic

Next, factor the quadratic expression inside the parentheses. We need to find two numbers that multiply to 15 and add to 8: $$m (m + 3)(m + 5) = 0.$$

Step 4 - Set Each Factor to Zero

Set each factor equal to zero to find the solutions for 'm': $$m = 0$$ $$m + 3 = 0$$ $$m + 5 = 0.$$

Step 5 - Solve for 'm'

Solve the equations from Step 4: $$m = 0$$ $$m = -3$$ $$m = -5.$$

Step 6: Conclusion

Combine the solutions to state the final answer: The solutions are: $$m = 0, -3, -5.$$

Key Concepts

Cubic Functions

Cubic functions are polynomial functions of degree three. The general form looks like this:
\(ax^3 + bx^2 + cx + d = 0\).
They are called 'cubic' because the highest power of the variable (commonly 'x') is three.
In our exercise, the equation \(m^3 + 8m^2 + 15m = 0\) is a cubic function.
Cubic functions can have one, two, or three real roots (solutions).
These are the points where the function intersects the x-axis.

Greatest Common Factor

To make solving polynomial equations easier, you often start by factoring out the Greatest Common Factor (GCF).
The GCF of a polynomial is the largest term (factor) that each term in the polynomial shares.
In the equation \(m^3 + 8m^2 + 15m = 0\), we see that each term has at least one 'm'.
So, we can factor 'm' out: \(m(m^2 + 8m + 15) = 0\).
This step simplifies the equation and reveals a quadratic equation in the parentheses.

Quadratic Equations

A quadratic equation is a polynomial of degree two, and its general form is \(ax^2 + bx + c = 0\).
In the original exercise, the quadratic equation inside the parentheses is \(m^2 + 8m + 15\).
To factor this quadratic equation, look for two numbers that multiply to 15 (the constant term) and add up to 8 (the coefficient of the linear term):
These numbers are 3 and 5 because \(3 * 5 = 15\) and \(3 + 5 = 8\).
Thus, the factored form is \((m + 3)(m + 5)\).
So, \(m(m^2 + 8m + 15)\) becomes \(m(m + 3)(m + 5) = 0\).

Solving Equations

After factoring the polynomial equation, the next step is solving for 'm'.
You set each factor equal to zero:

• \(m = 0\)
• \(m + 3 = 0\)
• \(m + 5 = 0\)

Solving these equations gives us the solutions:

• \(m = 0\)
• \(m = -3\)
• \(m = -5\)

Combining these, the solutions to the original cubic equation are \(m = 0, -3, -5\).
This means that these values are the roots of the cubic function, or the points where the function intersects the x-axis.