Problem 75 - Algebra and Trigonometry

Question

Factor the expression completely. $$ 2 x^{3}+4 x^{2}+x+2 $$

Step-by-Step Solution

Answer

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

Step 1: Group the Terms

First, group the terms in pairs: \[ (2x^3 + 4x^2) + (x + 2) \] This allows us to factor by grouping.

Step 2: Factor Common Terms from Each Group

Factor out the greatest common factor from each group. For the first pair:\[ 2x^2(x + 2) \] and for the second pair: \[ 1(x + 2) \] The expression becomes: \[ 2x^2(x + 2) + 1(x + 2) \]

Step 3: Factor Out the Common Binomial

Recognize that \((x + 2)\) is a common factor. Factor \((x + 2)\) from the entire expression:\[ (x + 2)(2x^2 + 1) \]

Step 4: Check for Further Factoring

Examine each factor to see if further factoring is possible. Both \(x + 2\) and \(2x^2 + 1\) are irreducible over the integers, so no further factoring is possible.

Key Concepts

Understanding Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and mathematical operations. They are the foundation of algebra and help us to describe mathematical relationships in a generalized form. For example, the expression \(2x^3 + 4x^2 + x + 2\) consists of terms which are combined using addition.
Each term can have a coefficient (a numerical factor), a variable (like \(x\)), and an exponent (showing the power to which the variable is raised). In \(2x^3\), the coefficient is 2, the variable is \(x\), and the exponent is 3. Understanding how to manipulate and simplify these expressions is crucial in mathematics education as it provides the skills necessary to solve more complex equations and problems.
In mathematics, clear comprehension of these expressions allows students to build on basic algebraic concepts and apply them in geometry, calculus, and even statistics. Recognizing patterns and structures within expressions is a critical skill that aids in the understanding and application of mathematical principles.

Exploring Polynomial Factorization

Polynomial factorization is the process of expressing a polynomial as a product of its factors. This method simplifies complex expressions, making them easier to work with and solve. When we factor the polynomial \(2x^3 + 4x^2 + x + 2\), we look for common terms or patterns.
The approach in factorization usually involves identifying and grouping terms that can be factored together, as seen in the original solution where terms were grouped into \((2x^3 + 4x^2) + (x + 2)\). Recognizing and factoring out common terms helps in reducing the polynomial into simpler expressions.

• The greatest common factor is extracted from each group.
• Identify shared factors across groups, like \((x + 2)\) in this instance.
• Use these shared factors to consolidate and simplify the expression.

This leads to the factored form \((x + 2)(2x^2 + 1)\), showcasing how complex expressions can often be broken down into more manageable components. Understanding this process is essential for solving quadratic equations and higher-degree polynomials in algebra.

Mathematics Education through Practice

Effective mathematics education emphasizes the importance of practice and application of concepts. Factorization, as seen in the polynomial equation example, is integral to developing problem-solving skills. By learning to factor polynomials, students enhance their analytical abilities and gain confidence in handling algebraic expressions.
Practicing various problems helps in recognizing different types of patterns and structures. It broadens understanding and reinforces the knowledge necessary to approach more challenging mathematical ideas. Teachers often provide diverse sets of problems to encourage this exploration and understanding.

• Consistency in practice helps solidify learning.
• Application of different problem-solving techniques improves adaptability.
• Constructive feedback aids in correcting misconceptions and refining skills.

Therefore, incorporating polynomial factorization into the broader curriculum helps students see its applications in various mathematical fields and real-world scenarios, fostering a deeper appreciation and competence in mathematics.