Problem 21 - Algebra A Combined Function

Question

Factor each trinomial completely. $$ m^{3}+18 m^{2}+81 m $$

Step-by-Step Solution

Answer

The trinomial is factored as \(m(m + 9)^{2}\).

Step 1: Identify the Common Factor

First, identify any common factors in the terms of the trinomial. The expression given is \(m^{3} + 18m^{2} + 81m\). Notice that each term has a factor of \(m\). Factoring \(m\) from each term gives us:\[m(m^{2} + 18m + 81)\]

Step 2: Factor the Quadratic Expression

Next, factor the quadratic expression \(m^{2} + 18m + 81\). Look for two numbers that multiply to \(81\) (the constant term) and add up to \(18\) (the coefficient of \(m\)). These numbers are \(9\) and \(9\) since \(9 \times 9 = 81\) and \(9 + 9 = 18\). Thus, the quadratic can be factored as:\[(m + 9)(m + 9)\] which is the same as \((m + 9)^{2}\).

Step 3: Write the Complete Factorization

Combine the factored quadratic back with the common factor from Step 1. The expression now fully factored is:\[m(m + 9)^{2}\].This represents the complete factorization of the original trinomial.

Key Concepts

Common Factor

A great start to factoring any trinomial is to find a common factor. In simple terms, a common factor is a number or variable that divides each part of the expression without leaving any leftovers. Think of it as a shared piece among all terms. For the expression \(m^{3} + 18m^{2} + 81m\), notice that each term includes the variable \(m\). This makes \(m\) a common factor.

• Extracting this common factor simplifies the expression, reducing it to \(m(m^{2} + 18m + 81)\).
• This critical first step makes working with the expression easier and sets the stage for further reduction.

Identifying and pulling out a common factor is essential before diving into more complex factoring techniques. It ensures that you are working with the simplest version of the expression possible before tackling the quadratic expression.

Quadratic Expression

Now, let's explore how to factor the quadratic expression \(m^{2} + 18m + 81\). This expression is in a standard quadratic form, which typically looks like \(ax^2 + bx + c\). Here, \(a = 1\), \(b = 18\), and \(c = 81\).

In factoring quadratics like these, the goal is to find two numbers that multiply to the constant term (\(c = 81\)) and add up to the coefficient of the linear term (\(b = 18\)).

• The factors we are looking for are both 9 because \(9 \times 9 = 81\) and \(9 + 9 = 18\).
• Thus, the expression can be written as \((m + 9)(m + 9)\), which is the same as \((m + 9)^{2}\).

Factoring quadratic expressions can initially seem challenging, but focusing on finding these specific pairs of numbers simplifies the process significantly.

Complete Factorization

After factoring the quadratic part, combine it back with the common factor. This gives the complete factorization of the original trinomial. For the expression \(m^{3} + 18m^{2} + 81m\), we already factored out \(m\) and reduced it to \((m + 9)^{2}\).
To arrive at the final solution, put the common factor with the factored quadratic:

• The complete factorization becomes \(m(m + 9)^{2}\).

This final form shows that you have successfully broken down the original trinomial into its most elemental pieces. It's vital in algebra because it reveals the roots and solutions that make the equation zero, offering deeper insight into its behavior. Understanding how complete factorization works can greatly enhance your problem-solving skills for more complex expressions in algebra.