Problem 63 - Algebra for College Students

Question

Factor by grouping. $$2 x^{2}+8 x+x+4$$

Step-by-Step Solution

Answer

The expression factors to \((2x + 1)(x + 4)\).

Step 1: Group Terms

Group the terms in pairs to prepare for factoring by grouping: \[ (2x^2 + 8x) + (x + 4) \]

Step 2: Factor Out Common Factors from Each Pair

First, factor out the greatest common factor from each pair of terms. For the first pair \(2x^2 + 8x\), the common factor is \(2x\): \[ 2x(x + 4) \]For the second pair \(x + 4\), there is no common factor other than 1, so it remains the same:\[ 1(x + 4) \]

Step 3: Factor Out the Common Binomial

Now factor out the common binomial \((x + 4)\) from both terms:\[ (2x + 1)(x + 4) \]

Step 4: Check the Work

Verify the factorization by expanding \((2x+1)(x+4)\) back to the original polynomial:\[ 2x(x+4) + 1(x+4) = 2x^2 + 8x + x + 4 = 2x^2 + 9x + 4 \]The original expression was misaligned in the calculation, we confirm that the steps need revisiting. However, the idea of factorization by grouping is correctly applied to the closest expression possible.

Key Concepts

Polynomial Expressions

Polynomial expressions are mathematical expressions that involve variables raised to whole number powers and their coefficients. These expressions can include addition, subtraction, and multiplication of variables and constants. For example, in the polynomial expression \(2x^2 + 8x + x + 4\), you'll notice that each term consists of a coefficient and a variable. The terms in a polynomial expression are usually ordered in descending powers of the variable, which makes calculations and simplifications easier.Polynomials can be classified by the number of terms they have:

• Monomials: A single term. Example: \(3x^2\).
• Binomials: Two terms. Example: \(x + 4\).
• Trinomials: Three terms. Example: \(x^2 + 5x + 6\).

Recognizing and understanding polynomial expressions is crucial when solving algebraic equations and factoring them, as it forms the basis for more advanced algebraic techniques.

Factoring by Grouping

Factoring by grouping is a method used to simplify polynomial expressions, typically those with four terms or more. It involves grouping terms with common factors and factoring them out. This technique is especially useful when there isn't an obvious common factor for the entire polynomial. Let's break down the steps as applied to our example:First, we split the polynomial \(2x^2 + 8x + x + 4\) into two groups: \((2x^2 + 8x)\) and \((x + 4)\). By grouping these terms, we prepare them for separate factorization.Next, we look for the greatest common factor in each group. In the first group \(2x^2 + 8x\), the common factor is \(2x\), which gives us \(2x(x + 4)\). For the second group \((x + 4)\), the common factor is 1, so it remains as is, \(1(x + 4)\).By recognizing the common binomial \((x + 4)\) in both groups, we can factor it out, resulting in \((2x + 1)(x + 4)\). This final expression is the factored form of the original polynomial, showing how effective this grouping method can be for simplifying and solving polynomial expressions.

Algebraic Techniques

Algebraic techniques are systematic methods used to simplify, manipulate, and solve polynomial equations and expressions. Factoring is one such technique that breaks down a complex equation into simpler, more manageable parts. This section will highlight the importance of selecting the correct algebraic technique and executing it accurately.In the process of factoring expressions, especially by grouping, you begin by:

• Identifying if grouping is viable: Useful for expressions typically with four terms.
• Determining the greatest common factor in each group to simplify the equation.
• Finding a common binomial factor in the groups once they are factored separately.

Proper application of algebraic techniques can greatly simplify problems initially viewed as daunting. Verification of your answer by expanding the factors back into the original polynomial is an essential step to confirm accuracy.For example, with our polynomial \((2x+1)(x+4)\), multiply the terms to check for correctness. This reversibility ensures the solutions are both correct, and more importantly, illustrate the power of algebraic techniques. With practice, these skills will become intuitive, streamlining your approach to more complex algebraic challenges.