Problem 9 - Elementary Algebra

Question

For the following problems, factor the trinomials when possible. $$ x^{2}+4 x+3 $$

Step-by-Step Solution

Answer

Answer: The factored form of the trinomial is \((x + 1)(x + 3)\).

Step 1: Write the Trinomial Expression

The given trinomial is: $$ x^2 + 4x + 3 $$

Step 2: Find Two Numbers that Multiply to 3 and Add to 4

We need to find two numbers that multiply to give 3 (the constant term), and add up to give 4 (the coefficient of the linear term). Let's consider the factors of 3: 1 * 3 = 3 -1 * -3 = 3 Among these pairs, the pair 1 and 3 also adds up to 1 + 3 = 4. So, we have found our two numbers: 1 and 3.

Step 3: Factor the Trinomial into Two Binomials

Now that we have the two numbers (1 and 3), we can factor our trinomial using these numbers: $$ x^2 + 4x + 3 = (x + 1)(x + 3) $$

Step 4: Final Answer

The factored form of the given trinomial is: $$ (x + 1)(x + 3) $$

Key Concepts

Algebraic Expressions

Algebraic expressions involve numbers, variables, and arithmetic operations like addition, subtraction, multiplication, and division. They are the foundation of algebra and are used to express relationships between quantities.
For example, in the expression \( x^2 + 4x + 3 \), we have:

• Variables: \(x\) appears in both the quadratic and linear terms.
• Coefficients: 4 is the coefficient of the linear term.
• Constant: 3 is a constant term.

Algebraic expressions can represent various scenarios, such as the area of a shape or the total cost of items. Understanding how to manipulate these expressions is key to solving equations and simplifying complex problems. To manage algebraic expressions, we often combine like terms, use the distributive property, and factor trinomials like in our exercise.

Polynomials

Polynomials are a specific type of algebraic expression consisting of variables and coefficients, combined using only addition, subtraction, and multiplication. They do not include division by variables. The degree of a polynomial is determined by the highest power of the variable in the expression.
In our given expression \( x^2 + 4x + 3 \), we have a:

• Quadratic term: \(x^2\), which indicates that this is a second-degree polynomial.
• Linear term: \(4x\), the first-degree part.
• Constant term: 3, the zero-degree part.

Polynomials can be factored, which involves finding expressions (usually binomials) that multiply to give the polynomial. This is a fundamental skill in algebra, as factoring is necessary for simplifying expressions and solving equations. Recognizing the structure of a polynomial helps us use appropriate strategies like finding factors that meet specific arithmetic conditions, such as in this exercise, efficiently.

Quadratic Equations

Quadratic equations are polynomials where the highest exponent of the variable is 2. These equations take the standard form \( ax^2 + bx + c = 0 \). Our trinomial expression \( x^2 + 4x + 3 \) is a perfect example when set to an equation form as \( x^2 + 4x + 3 = 0 \). To solve such equations, we often convert them from this expanded form into a factored form.
There are several methods to solve quadratic equations:

• Factoring: This involves expressing the quadratic as a product of two binomials. For example, we have \( (x + 1)(x + 3) = 0 \) in our solution, which shows the roots of the equation are \( x = -1 \) and \( x = -3 \).
• Completing the Square: This method manipulates the expression into a perfect square trinomial.
• Quadratic Formula: Useful when factoring is complicated or impossible, \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \).

Understanding how to navigate these methods allows students to solve real-world problems modeled by quadratic equations. Factoring, particularly, is a skill that provides insights into the properties of the problem at hand.