Problem 131 - Elementary Algebra

Question

Many trinomials of the form \(x^{2}+b x+c\) factor into the product of two binomials \((x+m)(x+n)\). Explain how you find the values of \(m\) and \(n\).

Step-by-Step Solution

Answer

Find integers \(m\) and \(n\) such that \(m \times n = c\) and \(m + n = b\). Then factor as \((x + m)(x + n)\).

Step 1 - Identify coefficients

Start by identifying the coefficients in the given trinomial. For a trinomial of the form \(x^2 + bx + c\), identify the values of \(b\) and \(c\).

Step 2 - Find factors of the constant term

Look for pairs of integers \((m, n)\) such that when multiplied, they produce the constant term \(c\).

Step 3 - Check the sum

Out of the pairs of integers found in Step 2, identify the pair \((m, n)\) whose sum equals the coefficient \(b\). This means solving \(m + n = b\).

Step 4 - Write the factors

Once the correct values of \(m\) and \(n\) are found, write the trinomial as the product of two binomials: \((x + m)(x + n)\).

Key Concepts

Trinomial

A trinomial is a type of polynomial which consists of exactly three terms. Polynomials are mathematical expressions containing variables and coefficients, combined using addition, subtraction, and multiplication. Specifically, a trinomial of the form \(x^{2} + bx + c\) includes:

• A squared term \(x^{2}\).
• A linear term \(bx\) where \(b\) is the coefficient.
• A constant term \(c\).

Such trinomials are common in quadratic equations, making it crucial to understand how to factor them for solving problems.

Binomial

A binomial is a polynomial with exactly two terms. Each term can be a constant, variable, or product of constants and variables. A typical binomial is represented as \(a + b\) or \(a - b\). When factoring a trinomial like \(x^{2} + bx + c\), the aim is to express it as the product of two binomials:
\((x + m)(x + n)\).
By expanding these binomials, i.e., multiplying them together, you can retrieve the original trinomial:
\(x^{2} + (m+n)x + mn\).
Understanding this relationship is the key to successful factoring.

Coefficients

Coefficients are the numerical factors that multiply the variables in polynomials. In a trinomial \(x^{2} + bx + c\), the coefficients are:

• 1 for the \(x^{2}\) term (implicitly).
• \(b\) for the \(bx\) term.
• \(c\) for the constant term.

When factoring trinomials, the coefficient \(b\) helps identify the sum of the integers \(m\) and \(n\), while the coefficient \(c\) directs us to their product. Hence, correctly identifying and leveraging these coefficients simplifies the factorization process.

Factoring

Factoring in algebra involves breaking down a complex expression into a product of simpler expressions. For trinomials, this means expressing \(x^{2} + bx + c\) as \((x + m)(x + n)\). Here’s a step-by-step approach:

• Identify \(b\) and \(c\) in the trinomial.
• Find pairs of integers \(m\) and \(n\) that multiply to \c\.
• Choose the pair where \(m + n = b\).
• Write the trinomial as \((x + m)(x + n)\).

By practicing this method, factoring trinomials becomes manageable, enabling easier problem-solving in algebra.