Problem 94 - Elementary Algebra

Question

Factor each trinomial of the form \(x^{2}+b x+c\). \(7 x+x^{2}+6\)

Step-by-Step Solution

Answer

\((x + 1)(x + 6)\)

Step 1: Rearrange the Trinomial

Rewrite the trinomial in standard form as \(x^2 + bx + c\). For this exercise, rearrange the terms of the trinomial: \(7x + x^2 + 6\) can be written as \(x^2 + 7x + 6\).

Step 2: Identify the Coefficients

Identify the coefficients \(b\) and \(c\) from the trinomial \(x^2 + 7x + 6\). In this case, \(b = 7\) and \(c = 6\).

Step 3: Find Factors of c

Find two numbers that multiply to \(c = 6\) and add up to \(b = 7\). These numbers are 1 and 6 because \(1 \times 6 = 6\) and \(1 + 6 = 7\).

Step 4: Write the Factored Form

Express the trinomial \(x^2 + 7x + 6\) as a product of two binomials using the numbers found: \( (x + 1)(x + 6)\).

Key Concepts

Quadratic Equations

A quadratic equation is any equation that can be arranged in the standard form: \[ ax^2 + bx + c = 0 \]. Here, \(x\) represents the variable, and \(a\), \(b\), and \(c\) are coefficients where \(a ≠ 0\). Quadratic equations are important in algebra because they provide a foundation for more complex topics like polynomial functions and calculus. They often describe a wide array of real-world phenomena such as projectile motion and area calculations. In our example equation, \(x^2 + 7x + 6 = 0\), we set the coefficients as \(a = 1\), \(b = 7\), and \(c = 6\). Understanding how these pieces fit together is the key to successfully solving quadratic equations.

Algebraic Expressions

An algebraic expression is a mathematical phrase that can include numbers, variables (like \(x\)), and operators (like \(+\) or \(-\)). For instance, in the expression \(x^2 + 7x + 6\), we have:

• \(x^2\) which is a quadratic term
• \(7x\) which is a linear term
• \(6\) which is a constant term

To manipulate these expressions, we perform operations such as addition, subtraction, and factoring. Factoring is particularly useful because it breaks down complex expressions into simpler, more workable parts. Doing so makes it easier to solve for the variable.

Polynomial Factoring

Factoring a polynomial means breaking it down into simpler polynomials that, when multiplied together, give us the original polynomial. In this case, our target is the trinomial \(x^2 + 7x + 6\). The first step is to arrange this trinomial in standard form as \(x^2 + 7x + 6\). Next, we identify the coefficients \(b\) and \(c\), which are 7 and 6, respectively. After identifying these coefficients, we find the two numbers that multiply to \(c\) (which is 6) and add up to \(b\) (which is 7). These numbers are 1 and 6. Thus, we can write our trinomial as:
\((x + 1)(x + 6)\). This is a crucial skill in elementary algebra and is foundational for solving more advanced polynomial equations.

Elementary Algebra

Elementary Algebra covers basic algebraic concepts, which include operations with algebraic expressions, solving equations, and understanding properties of numbers. In this context, factoring trinomials like \(x^2 + 7x + 6\) is a fundamental skill. Here's a brief overview of the process:

• Rearrange the trinomial into the standard form \(x^2 + bx + c\).
• Identify the coefficients \(b\) and \(c\). For our example, these are 7 and 6.
• Find two numbers that multiply to \(c\) and add up to \(b\). For our example, these numbers are 1 and 6.
• Express the trinomial as the product of two binomials: \((x + 1)(x + 6)\).

Mastering these basics can make tackling more complex algebraic problems significantly easier.