Problem 170 - Elementary Algebra

Question

Factor. \(3 m^{2}+26 m+48\)

Step-by-Step Solution

Answer

(3m + 8)(m + 6)

Step 1 - Identify Coefficients

Identify the coefficients for the quadratic equation. Here, the coefficients are: a = 3, b = 26, and c = 48.

Step 2 - Find the Product ac

Calculate the product of the coefficients a and c: ac = 3 * 48 = 144.

Step 3 - Find Two Numbers that Multiply to ac and Add to b

Identify two numbers that multiply to 144 and add to 26. These numbers are 18 and 8 because: 18 * 8 = 144 and 18 + 8 = 26.

Step 4 - Rewrite the Middle Term

Rewrite the middle term (26m) using the two numbers found in step 3: 3m^2 + 18m + 8m + 48.

Step 5 - Factor by Grouping

Group the terms to factor by grouping: (3m^2 + 18m) + (8m + 48).

Step 6 - Factor Out the Greatest Common Factor

Factor out the greatest common factor from each group: 3m(m + 6) + 8(m + 6).

Step 7 - Factor Out the Common Binomial Factor

Factor out the common binomial factor (m + 6): (3m + 8)(m + 6).

Key Concepts

Quadratic Equations

A quadratic equation is a polynomial equation of degree 2, usually in the form ax^2 + bx + c = 0. Here, 'a', 'b', and 'c' are coefficients, and x is the variable. Quadratic equations often appear in various problems, and solving them can help determine the values of x that satisfy the equation.
There are different methods to solve quadratic equations, including:

• Factoring
• Using the quadratic formula
• Completing the square

In this specific exercise, the equation given is in the form of a quadratic expression, not set to zero. The goal here is to factor the expression instead of finding the roots. Factoring quadratics is critical as it allows us to simplify the expression or solve equations more easily later.

Factoring by Grouping

Factoring by grouping is a technique used to factor more complex polynomials. By splitting the middle term and grouping, we can simplify the expression into factors. This method is particularly useful when dealing with quadratics that are not easily factorable in one step.
To factor by grouping, follow these steps:

• Identify the coefficients and calculate the product ac.
• Find two numbers that multiply to ac and add to b.
• Rewrite the middle term using these two numbers.
• Group the terms into pairs and factor out the greatest common factor from each pair.
• Factor out the common binomial factor.

Being thorough with your step-by-step process ensures you don’t miss any simplification opportunities and correctly factor the polynomial.

Greatest Common Factor

The Greatest Common Factor (GCF) is the largest number or variable expression that divides all terms in a polynomial without leaving a remainder. Identifying the GCF is a crucial step when factoring polynomials, especially when using methods like factoring by grouping.
In our example, we found the GCF for the grouped terms:

• In the first group, (3m^2 + 18m), the GCF is 3m. Factoring it out gives us 3m(m + 6).
• In the second group, (8m + 48), the GCF is 8. Factoring it out gives us 8(m + 6).

After factoring out the GCF from each group, both groups had a common binomial factor (m + 6), which was then factored out to achieve the final factored form (3m + 8)(m + 6).
Always ensure to check each group separately and correctly find the GCF. This skill is essential for simplifying polynomials correctly and efficiently.