Problem 103 - Algebra 1: Concepts and Skills

Question

Factor the trinomial. (Lessons 10.5,10.6) $$ x^{2}+4 x+4 $$

Step-by-Step Solution

Answer

The factored form of the trinomial \(x^2 + 4x + 4\) is \((x+2)^2\).

Step 1: Identify the coefficients

First we need to identify the coefficients in the given trinomial expression, in particular \(b\) and \(c\) for the general quadratic \(x^2 + bx + c\). Here, \(b=4\) and \(c=4\).

Step 2: Identify factors

Now, find two numbers that multiply to \(c=4\) and add to \(b=4\). These two numbers are 2 and 2, since \(2 * 2 = 4\) and \(2 + 2 = 4\).

Step 3: Write the factorized form

Finally, now we write the trinomial in its factorized form using these identified numbers. The factored form of the given trinomial is \((x+2)(x+2)\) or equivalently, \((x+2)^2\).

Key Concepts

Quadratic Equations

Understanding quadratic equations is pivotal for solving a multitude of algebraic problems, including the factorization of trinomials. Quadratic equations can be recognized by their standard form, which is \( ax^2+bx+c=0 \), where \( a \), \( b \) and \( c \) are coefficients, and \( a \) is not equal to zero. The solutions to these equations, also referred to as the roots, can often be found by factoring when the equation is set to zero. Factoring is a process of breaking down the equation into simpler expressions that can be multiplied to give the original equation.

In the given exercise: \( x^2+4x+4=0 \), we identify it as a quadratic equation by its structure. There are various methods of solving quadratic equations, such as completing the square, using the quadratic formula, or factorization, which is discussed in our exercise. To successfully navigate through the complexities of quadratic equations, grasping the concept of factorization is essential and provides a foundation for students to solve these types of equations efficiently.

Factoring Algebraic Expressions

Factoring algebraic expressions, especially polynomials, is an essential skill. This technique involves expressing an algebraic expression as a product of its factors, much like breaking down a number into a product of prime numbers. Simple trinomials, like \( x^2 + bx + c \), can often be factored into binomial terms. The process typically involves finding two numbers that both add up to \( b \) and multiply to \( c \) when \( a \) (the coefficient of \( x^2 \)) equals 1.

The exercise provided involves factoring the simple trinomial \( x^2+4x+4 \) which, as we can see in the step by step solution, factored neatly into \( (x+2)^2 \). This square form indicates that the two factors of the trinomial are identical. By mastering the process of factoring, students can solve quadratic equations more easily and understand the structure of algebraic expressions in greater depth.

Coefficient Identification

Coefficient identification is the initial step in managing algebraic expressions and equations. Coefficients are the numerical or constant part of the terms in an algebraic expression. In a quadratic equation of the form \( ax^2 + bx + c = 0 \), \( a \) is the coefficient of the quadratic term, \( b \) is the coefficient of the linear term, and \( c \) is the constant term. Identifying these coefficeints accurately is crucial to factor the trinomial correctly.

In the given exercise, \( b \) and \( c \) were identified as 4 and 4, respectively, from the expression \( x^2 + 4x + 4 \). With the correct identification of these coefficients, one can determine the suitable factors that can be used to re-write the trinomial as a product of binomials. This makes it easier to proceed with factoring the algebraic expression and is an important skill to solve not only trinomials but a wide range of algebraic problems.