Question

Factor each polynomial. $$ x^{2}+5 x+6 $$

Short answer

The factored form is \((x + 2)(x + 3)\).

Step-by-step solution

Identify the Polynomial

First, look at the given polynomial: \(x^2 + 5x + 6\)

Find Two Numbers That Multiply and Add

Find two numbers that multiply to the constant term (6) and add to the coefficient of the middle term (5). The numbers are 2 and 3 since \(2 \times 3 = 6\) and \(2 + 3 = 5\).

Write the Polynomial in Factored Form

Using the numbers found, write the polynomial as a product of two binomials: \((x + 2)(x + 3)\).

Verify the Factoring

Expand \((x + 2)(x + 3)\) to ensure it equals the original polynomial: \((x + 2)(x + 3) = x^2 + 3x + 2x + 6 = x^2 + 5x + 6\). This confirms the factoring is correct.

Key concepts

Polynomial Factorization

Polynomial factorization is a fundamental concept in algebra. It involves expressing a polynomial as the product of simpler polynomials. For example, we factorized the polynomial \(x^2 + 5x + 6\) by expressing it as \((x + 2)(x + 3)\). This not only simplifies the polynomial but also helps to solve equations more easily. The main goal is to break down the polynomial into products of binomials or other polynomials that are easier to manage.

Quadratic Equations

Quadratic equations are another critical aspect of algebra. They are polynomial equations of degree 2, generally written in the form \(ax^2 + bx + c = 0\). In the exercise, we looked at the quadratic equation \(x^2 + 5x + 6\). One common method to solve these is by factoring. Once we have factored the quadratic equation, we can set each factor equal to zero to find the solutions of the equation. So, for our polynomial, once factored to \((x + 2)(x + 3)\), setting each part to zero gives the solutions \(x = -2\) and \(x = -3\).

Algebraic Methods

Algebraic methods refer to various techniques used to simplify, factor, and solve algebraic expressions and equations. In this exercise, we used simple factoring, a fundamental algebraic method. The steps included:

• Identifying the polynomial
• Finding two numbers that multiply to the constant term and add to the middle coefficient
• Writing the polynomial in its factored form
• Verifying the factorization by expanding the factors

These steps are part of a larger toolkit that includes completing the square, using the quadratic formula, and more. The choice of method usually depends on the specific problem and its complexity.