Problem 75 - Elementary Algebra

Question

Factor each trinomial of the form \(x^{2}+b x+c\). \(x^{2}-8 x+12\)

Step-by-Step Solution

Answer

The trinomial \(x^{2} - 8x + 12\) factors to \((x - 2)(x - 6)\).

Step 1 - Identify the Coefficients

Observe the given trinomial in the form \(x^{2} + bx + c\). Here, \(b\) is the coefficient of the linear term and \(c\) is the constant term. For the trinomial \(x^{2} - 8x + 12\), identify \(b = -8\) and \(c = 12\).

Step 2 - Find Two Numbers That Multiply to c and Add to b

Look for two numbers that multiply to \(c = 12\) and add up to \(b = -8\). These numbers are \(-2\) and \(-6\) because \((-2) \times (-6) = 12\) and \((-2) + (-6) = -8\).

Step 3 - Write the Factors

Rewrite the trinomial as a product of two binomials using the numbers found in Step 2. Thus, \(x^{2} - 8x + 12\) factors to \((x - 2)(x - 6)\).

Key Concepts

Quadratic Equations

A quadratic equation is a type of polynomial equation of the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants. The quadratic equation is said to be in standard form when it is arranged in descending order of the powers of \(x\).

In the given exercise, we have a quadratic equation in the form \(x^2 - 8x + 12\).

To solve or factorize a quadratic equation, there are several methods like:

• Factoring
• Completing the Square
• Using the Quadratic Formula

The goal is to express the quadratic as a product of its linear factors. This exercise focuses on the factoring method. When the quadratic is put in the pattern \(x^2 + bx + c\), it becomes easier to find the factors directly.

Algebraic Factoring

Algebraic factoring is the process of breaking down a polynomial into products of simpler polynomials. This technique is essential in solving polynomial equations and simplifying expressions.

In the exercise provided, we used algebraic factoring to factor the quadratic trinomial \(x^2 - 8x + 12\). Here’s how the steps break down:

• **Identify the Coefficients:** Recognize \(b\) and \(c\) from the trinomial \(x^2 + bx + c\).
• **Find Two Numbers:** Look for two numbers that multiply to the constant term \(c\) and add up to the linear coefficient \(b\).
• **Write the Factors:** Rewrite the trinomial as the product of two binomials using the numbers found.

These steps help break down or 'factor' a polynomial into simpler binomial parts. This is particularly useful for solving equations by setting each factor to zero and solving for \(x\).

Polynomials

A polynomial is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients. Polynomials can take on various forms, including monomials, binomials, and trinomials, depending on the number of terms involved.

For example, in the exercise, we are dealing with a trinomial, which is a polynomial with three terms: \(x^2\), \(-8x\), and \(+12\). The concept of factoring polynomials is crucial because it simplifies these expressions and helps solve equations more effectively.

Here are some key points about polynomials:

• **Polynomial Degree:** The degree of a polynomial is the highest power of the variable in the expression.
• **Standard Form:** Polynomials are usually written in standard form, with terms in descending order of exponents.
• **Factoring:** Factoring converts a polynomial into a product of simpler polynomials, making it easier to analyze and solve.

Understanding and working with polynomials form the backbone of algebra, and mastering this skill is essential for progressing in mathematics. By practicing factoring, students get better at simplifying complex problems and solving them efficiently.