Problem 11 - Introductory Algebra for College Students

Question

Before getting to multiple-step factorizations, let's be sure that you are comfortable with exercises requiring only one of the factoring techniques. Factor each polynomial. $$x^{2}-2 x+1$$

Step-by-Step Solution

Answer

The factorization of the polynomial \(x^{2} - 2x + 1\) is \((x - 1)^2\).

Step 1: Identify the Quadratic Form

The polynomial \(x^{2} - 2x + 1\) has the quadratic form \(x^{2} - 2ab + b^{2}\). We can make an analogy from this form where \(a=x\) and \(b=1\). This is proven by fact that 'a' or \(x\) is squared, -2 times 'a' times 'b' or -2*x*1 = -2x, and 'b' squared or \(1^2 = 1\).

Step 2: Use the Factoring Formula

Now we know the values of 'a' and 'b'. Applying these values to the formula \((a - b)^2\) will give us the factored form of the polynomial. Substituting \(a\) with x and \(b\) with 1, the polynomial is factored into \((x - 1)^2\).

Key Concepts

Polynomial Factorization

Polynomial factorization is a process used to express a polynomial as the product of its factors. This means we break down a complex expression into simpler, more manageable parts or expressions that, when multiplied together, produce the original polynomial. Understanding polynomial factorization is crucial because it simplifies many algebraic expressions and makes solving equations more straightforward.

• Polynomials can be factorized in various ways depending on their structure. For instance, the expression \(x^2 - 2x + 1\) is a trinomial, which means it consists of three terms that can often be broken into factors that are binomials, like \((x-1)(x-1)\).
• Factoring helps in solving equations since it can transform a polynomial equation into a set of simpler equations that are easier to handle. Recognizing patterns like perfect square trinomials is essential in polynomial factorization.

Studying polynomial factorization allows students to see deeper patterns within algebra, and it lays the foundation for solving more complex algebraic equations.

Factoring Techniques

There are several factoring techniques used to break down algebraic expressions, especially polynomials. Here are some of the most common techniques that students learn:

• Factoring by grouping: This technique involves grouping terms with common factors and then factoring them out. It's particularly useful for polynomials where terms do not readily factor as in simple quadratic or binomial patterns.
• Factoring trinomials: Many quadratic expressions can be factored using patterns, such as recognizing a perfect square trinomial like \(x^2 - 2x + 1\), which factors to \((x-1)^2\). Knowing such patterns allows students to factor quickly.

Applying the correct factoring technique depends on the form of the polynomial at hand. Mastery of these techniques provides students tools to simplify expressions and solve equations efficiently.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and arithmetic operations (like addition, subtraction, multiplication, and division). These expressions form the basis of algebra and include polynomials as a major subset. Understanding these expressions is critical to factoring them effectively.
Polynomials, which are algebraic expressions themselves, require careful consideration when factoring:

• Identifying common factors: When given a polynomial expression, look for terms that share factors. This makes the expression easier to factor.
• Simplifying expressions: Breaking down complex polynomial expressions into simpler pieces through factorization, helps in solving it or understanding its properties.

Grasping the fundamentals of algebraic expressions and how to manipulate them paves the way for more advanced mathematical problem-solving and helps students in transitioning into higher-level algebra seamlessly.