Problem 33 - Algebra and Trigonometry

Question

29-34 . Factor the expression by grouping terms. $$ x^{3}+x^{2}+x+1 $$

Step-by-Step Solution

Answer

The expression is factored as \((x + 1)(x^2 + 1)\).

Step 1: Group Terms

The given expression is \(x^3 + x^2 + x + 1\). To factor this by grouping, we first separate the expression into two groups: \((x^3 + x^2) + (x + 1)\).

Step 2: Factor Out Common Factors in Each Group

Look at each group separately. In the first group \((x^3 + x^2)\), factor out \(x^2\) to get \(x^2(x + 1)\). In the second group \((x + 1)\), the expression is already a common factor, \((x + 1)\). Now the expression is \(x^2(x + 1) + 1(x + 1)\).

Step 3: Factor Out the Common Binomial

Notice that \((x + 1)\) is a common factor in both groups. Factor \((x + 1)\) out of the entire expression to get \((x + 1)(x^2 + 1)\).

Key Concepts

Understanding Algebraic Expressions

Algebra is like a language, and algebraic expressions are its sentences. An algebraic expression is a combination of numbers, variables, and operations like addition, subtraction, multiplication, and division. In our example, the expression is given as \(x^3 + x^2 + x + 1\).

Each part of the expression separated by a plus or minus sign is known as a 'term'. Here, \(x^3\), \(x^2\), \(x\), and \(1\) are the terms of the expression.

• Terms with variables raised to a power, like \(x^3\) and \(x^2\), are called polynomial terms.
• Terms without variables, like \(1\), are called constants.

Working with algebraic expressions often involves recognizing patterns or grouping terms that can simplify the expression. By mastering this, complex expressions can be reduced or factored to a more manageable form. Thus, understanding the breakdown of these expressions is crucial.

The Grouping Method

The grouping method is a neat technique that can break down complex polynomials into simpler factors, especially when it lacks easy identifications like common monomials. This method involves organizing the terms into groups that share a common factor.

In the step-by-step solution provided, the expression \(x^3 + x^2 + x + 1\) is divided into two groups: \((x^3 + x^2)\) and \((x + 1)\). Both groups should preferably have a common factor that can be factored out, which eventually makes the expression simpler.

Here’s how it helps:

• It makes complex expressions more manageable by highlighting underlying patterns.
• Reduces errors compared to trial-and-error methods of factoring.
• Saves time by converting the expression into a product of factors directly.

Remember, the key is to look for commonality in each group you create. Once this is factored out, you can further simplify and solve the polynomial equation.

Working with Polynomial Equations

Polynomial equations consist of variables raised to positive integer powers with constant coefficients. These equations can be classified based on the degree of the polynomial, which is defined by the highest power of the variable present. In our example, \(x^3 + x^2 + x + 1\) is a cubic polynomial equation because the highest degree is three.

Factoring polynomial equations involves breaking them down into the product of simpler polynomials, which makes solving these equations much simpler. For example, the expression was restructured using the grouping method to \((x + 1)(x^2 + 1)\).

When tackling polynomial equations, understanding the following is critical:

• The degree indicates the number of potential solutions or roots the equation can have.
• Factored form gives insight into the roots and makes it easier to solve the equation.
• Factoring transforms complex equations into a set of simpler equations that can be solved independently.

Therefore, learning how to manipulate and factor polynomial equations is an essential skill in algebra, as it forms the foundation for more advanced topics and applications.