Problem 12 - Elementary Algebra

Question

Factor. $$ x 2+4 x+3 $$

Step-by-Step Solution

Answer

The factored form is \((x + 1)(x + 3)\).

Step 1: Identify the Polynomial

The given polynomial is a quadratic equation in the form of \( ax^2 + bx + c \). Here, \( a = 1 \), \( b = 4 \), and \( c = 3 \).

Step 2: Confirm Leading Coefficient

Notice that the leading coefficient \( a = 1 \), which simplifies the factoring process as we focus primarily on the middle term and constant.

Step 3: Product-Sum Method

To factor the quadratic \( x^2 + 4x + 3 \), find two numbers that multiply to \( ac \) (which is 3) and add up to \( b \) (which is 4). These numbers are 1 and 3.

Step 4: Rewrite the Middle Term

Rewrite the expression \( x^2 + 4x + 3 \) using the numbers found: \( x^2 + 1x + 3x + 3 \).

Step 5: Group Terms

Group the expression into two pairs: \((x^2 + 1x) + (3x + 3)\).

Step 6: Factor Each Group

Factor out the greatest common factor from each group: \(x(x + 1) + 3(x + 1)\).

Step 7: Apply the Distributive Property

Notice that \((x + 1)\) is a common factor, factor it out: \((x + 1)(x + 3)\).

Step 8: Verify Factored Form

Multiply \((x + 1)(x + 3)\) to verify: \((x + 1)(x + 3) = x^2 + 3x + x + 3 = x^2 + 4x + 3\), confirming the factorization is correct.

Key Concepts

Product-Sum Method

The product-sum method is an efficient strategy used to factor quadratic expressions of the form \( ax^2 + bx + c \). This method is especially straightforward when the leading coefficient, \( a \), is 1. The core idea is to find two numbers that multiply to give the product of \( a \) and \( c \), and simultaneously add up to \( b \).

• Identify the coefficients: In our example, \( a = 1 \), \( b = 4 \), and \( c = 3 \).
• Calculate \( ac \), which in this case is \( 1 \times 3 = 3 \).
• Find two numbers that multiply to 3 and add to 4. These numbers are 1 and 3.

Once these numbers are determined, the quadratic expression can be rewritten by splitting the middle term based on the identified numbers. This step transforms the quadratic into a format that can be grouped, paving the way for factorization. Understanding this method is crucial as it forms the foundation for more complex factoring techniques.

Polynomial Equations

Polynomial equations encompass expressions consisting of variables and coefficients involving only the operations of addition, subtraction, multiplication, and non-negative integer exponents.

• The degree of a polynomial refers to the highest exponent; for example, \( x^2 + 4x + 3 \) is a quadratic polynomial because its highest power is 2.
• In solving polynomial equations, particularly quadratics, factorization is a favored technique as it simplifies the expressions into smaller, manageable parts.
• Each of these smaller parts, or factors, can be set to zero to find solutions to the equation, also known as 'roots'.

Quadratic equations, the most basic form of polynomials, often appear in various real-world scenarios, making this basic factorization skill extremely valuable. Grasping the nuances of polynomial behavior through their equations assists in exploring vast mathematical landscapes further down the road.

Distributive Property

The distributive property is a fundamental principle in algebra which allows you to expand or simplify expressions. It states that \( a(b + c) = ab + ac \). This property is viable both "backward" in expanding and "forward" in factoring.In the context of factoring quadratics, the distributive property enables us to factor out common terms from an expression.

• When we broke down the expression \((x^2 + 1x) + (3x + 3)\) into \(x(x + 1) + 3(x + 1)\), the property helped us recognize \((x + 1)\) as a common factor.
• We applied the distributive property "backwards" to factor \((x + 1)\) from both grouped terms, resulting in the product \((x + 1)(x + 3)\).
• This demonstration shows that recognizing common factors first can ease the complexity of a polynomial.

Mastering the distributive property is essential for simplifying expressions and solving equations effectively. It's a versatile tool that simplifies intricate algebraic manipulations and is an integral part of learning how to solve polynomial equations through factorization.