Problem 45 - Introductory Algebra for College Students

Question

Factor any perfect square trinomials, or state that the polynomial is prime. $$x^{2}+2 x+1$$

Step-by-Step Solution

Answer

This polynomial \(x^{2} + 2x + 1\) is a perfect square trinomial and it can be factored as \((x+1)^{2}\).

Step 1: Identify the form

The equation is a quadratic equation of the form \(x^{2}+2x+1\). Now, try to match this form to a perfect square trinomial, which is \(a^{2} + 2ab + b^{2}\).

Step 2: Identify 'a' and 'b'

In our equation, \(a = x\) and \(b=1\) because \(a^{2} = x^{2}\) and \(b^{2} = 1\).

Step 3: Express the polynomial as (a+b)^2

Now, express the polynomial in terms of \((a+b)^{2}\). So, \(x^{2}+2x+1\) can be written as \((x+1)^{2}\).

Key Concepts

Quadratic Equations

Quadratic equations are an integral part of algebra that involve an expression of the form ax^2 + bx + c = 0, where a, b, and c are constants and a is not equal to 0. The solutions to these equations are the values of x that make the equation true and are found using methods such as factoring, completing the square, or the quadratic formula.

Factoring a quadratic is one of the most essential skills in solving these equations. In the case of perfect square trinomials like x^2 + 2x + 1, factoring simplifies the problem significantly. Recognizing this specific form allows us to express the trinomial as a square of a binomial, making it much easier to solve for x by taking the square root of both sides.

Understanding how to identify and factor perfect square trinomials is crucial as it is not only a simple and quick method but it also helps in simplifying more complex problems involving quadratic equations.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations that form the building blocks of algebra. They can include terms, coefficients, constants, variables, and exponents. When we deal with expressions like x^2 + 2x + 1, we're seeing a polynomial - an expression that can have multiple terms connected by addition or subtraction.

In the context of factoring, comprehending the structure of these expressions is key to simplifying and solving equations. A perfect square trinomial is a specific type of algebraic expression where the first and last terms are perfect squares and the middle term is twice the product of the square roots of these perfect squares. Recognizing and manipulating these expressions allows us to perform operations such as factoring which greatly aids in solving equations.

By improving your understanding of algebraic expressions, especially how to identify and factor perfect squares, you gain a powerful tool in tackling a wide range of algebra problems.

Polynomial Factorization

Polynomial factorization is the process of breaking down a polynomial into a product of simpler polynomials. Similar to factoring numbers into primes, polynomial factorization simplifies expressions, making it easier to solve equations or perform other algebraic operations.

For instance, the perfect square trinomial x^2 + 2x + 1 factors neatly into (x + 1)^2, reducing the polynomial to a simpler form. Identifying this pattern is a handy skill, as these expressions frequently appear in algebra, and being able to quickly and accurately factor them can save time and reduce errors in computations and problem-solving.

Learning different factoring techniques for polynomials, such as factoring by grouping, using the difference of squares, or spotting a perfect square trinomial, provides a strong foundation to confidently navigate through complex algebraic problems.