Problem 84 - Elementary Algebra

Question

Factor by grouping. $$ 2 x 3+2 x 2-x-1 $$

Step-by-Step Solution

Answer

The factored form is \((x + 1)(2x^2 - 1)\).

Step 1: Understand the Problem

We need to factor the expression \(2x^3 + 2x^2 - x - 1\) by grouping terms that can be factored separately. This method will help us rewrite the expression as a product of simpler polynomials.

Step 2: Group Terms

Break down the expression into two groups: \((2x^3 + 2x^2)\) and \((-x - 1)\). This is the initial attempt to see if there's a common factor in each group.

Step 3: Factor Common Factors in Each Group

In the first group \( (2x^3 + 2x^2) \), factor out the common factor \(2x^2\), resulting in \(2x^2(x + 1)\). The second group \((-x - 1)\) has a common factor of \(-1\), so factoring it gives us \(-1(x + 1)\).

Step 4: Factor Out the Common Binomial Factor

Now, observe that both terms \(2x^2(x + 1)\) and \(-1(x + 1)\) contain the common binomial factor \((x + 1)\). Factor \((x + 1)\) out of the entire expression. This gives:\[ (x + 1)(2x^2 - 1) \].

Step 5: Verify and Simplify

Multiply the factors \((x + 1)(2x^2 - 1)\) to ensure it results in the original expression. Confirming this verifies that our factorization is correct.

Key Concepts

Polynomials

Polynomials are mathematical expressions that consist of variables and coefficients, combined using addition, subtraction, and multiplication. Each term in a polynomial is made up of a coefficient (a numerical factor), a variable (like \(x\)), and an exponent. For example, in the polynomial \(2x^3 + 2x^2 - x - 1\), each part is a term of the polynomial:

• \(2x^3\) - The coefficient is 2, the variable is \(x\), and the exponent is 3.
• \(2x^2\) - Coefficient is 2, variable is \(x\), exponent is 2.
• \(-x\) - Here, the coefficient is -1.
• \(-1\) - A constant term with no variable.

Polynomials can be categorized by their degree, which is determined by the highest exponent in the expression. In this case, the degree is 3, making it a cubic polynomial. Understanding polynomials is essential as they form the building blocks of more complex algebraic expressions.

Common Factors

The concept of common factors is key in simplifying polynomial expressions through factoring. A common factor in a group of terms is an expression that divides each term without leaving a remainder. For example, in the group \(2x^3 + 2x^2\), the common factor is \(2x^2\), because:

• \(2x^3 = 2x^2 \times x\)
• \(2x^2 = 2x^2 \times 1\)

Factoring the common factor simplifies the expression to \(2x^2(x + 1)\).

In the second group, \(-x - 1\), the common factor is -1, leaving us with \(-1(x + 1)\). Factoring out these common elements makes the expression easier to understand and manipulate, as seen in the factorization process that combines both groups.

Binomial Factors

Binomial factors are expressions with two terms connected either by addition or subtraction. In factoring by grouping, identifying a common binomial factor is crucial. For this problem, once each group is factored:

• First group: \(2x^2(x + 1)\)
• Second group: \(-1(x + 1)\)

Notice that both expressions contain the binomial \((x + 1)\). When a common binomial factor is present, it can be factored out from the expression. So, in this case, factoring out \((x + 1)\) gives us the final expression:\[(x + 1)(2x^2 - 1)\]

Finding and using binomial factors simplifies complex polynomial expressions, making them easier to handle and solve.