Problem 19 - Elementary and Intermediate Algebra

Question

Factor each trinomial. See Example 2 or Example \(10 .\) $$m^{2}-5 m+6$$

Step-by-Step Solution

Answer

The factored form of the trinomial is \((m - 2)(m - 3)\).

Step 1: Identify a, b, and c in the trinomial

The given trinomial is in the form of \( ax^2 + bx + c \). Here, \( a = 1 \), \( b = -5 \), and \( c = 6 \).

Step 2: Find two numbers that multiply to ac and add to b

Since \( a = 1 \), we need two numbers that multiply to \( 6 \) (\( a \times c = 6 \)) and add to \( -5 \) (\( b = -5 \)). The numbers that satisfy these conditions are \(-2\) and \(-3\).

Step 3: Rewrite the middle term using the numbers found

Express \(-5m\) as \(-2m - 3m\) to rewrite the trinomial: \( m^2 - 2m - 3m + 6 \).

Step 4: Factor by grouping

Group the terms as \((m^2 - 2m) + (-3m + 6)\). Factor out the greatest common factor from each group: \( m(m - 2) - 3(m - 2) \).

Step 5: Factor out the common binomial factor

Both groups contain the factor \((m - 2)\). Factor this out: \((m - 2)(m - 3)\). This is the factored form of the original trinomial.

Key Concepts

Polynomial Expressions

Polynomial expressions are like building blocks in algebra, comprising variables, coefficients, and exponents combined through addition, subtraction, or multiplication. A trinomial is a specific type of polynomial expression that has exactly three terms. In the expression \(m^2 - 5m + 6\), each part or term contributes to the structure of the polynomial:

• The first term, \(m^2\), is the quadratic term, and it dictates the degree of the trinomial, which is 2.
• The second term, \(-5m\), is linear and contributes significantly towards our factoring efforts, particularly through middle term splitting.
• The third term, \(6\), is a constant.

Understanding polynomial expressions is crucial for mastering algebraic operations, such as factoring, which simplify equations and make problem-solving more manageable.

Algebraic Factoring

Algebraic factoring involves breaking down a complex expression into simpler components or 'factors' that can easily multiply to form the original expression. It simplifies polynomials, making it easier to solve equations or analyze functions.For our trinomial \(m^2 - 5m + 6\), we use factoring to express it as a product of two simpler binomials:

• First, we identify coefficients, where \(a = 1\), \(b = -5\), and \(c = 6\).
• Then, find two numbers that multiply to \(a \times c\) and add to \(b\). In this case, they are \(-2\) and \(-3\).
• Finally, use these numbers to rewrite and factor the trinomial: \((m - 2)(m - 3)\).

Algebraic factoring not only aids in academic settings but is also a vital tool in various scientific and engineering applications.

Middle Term Splitting

Middle term splitting is an insightful technique used in factoring trinomials, particularly when factoring expressions where the coefficient of the leading term is 1 (as in our example). This method involves rewriting the middle term of the expression so it can be grouped with the other terms. Here's how it works for \(m^2 - 5m + 6\):

• Identify two numbers that combine to form the middle coefficient \(b = -5\) and multiply to \(c = 6\). Here, \(-2\) and \(-3\) work perfectly.
• Rewrite the expression by splitting the middle term using these numbers: \(m^2 - 2m - 3m + 6\).
• Group the terms into pairs: \((m^2 - 2m) + (-3m + 6)\), simplifying each pair to factor them individually before reaching the final factored form.

Middle term splitting makes apparent the hidden structure of a trinomial, revealing its factors step by step and emphasizing the elegance of algebraic manipulation.