Problem 30 - Algebra and Trigonometry

Question

29-34 . Factor the expression by grouping terms. $$ 3 x^{3}-x^{2}+6 x-2 $$

Step-by-Step Solution

Answer

The expression factors to \((3x - 1)(x^2 + 2)\).

Step 1: Identify Pairs

To factor by grouping, we first divide the expression into two pairs of terms. Here, we can group \(3x^3 - x^2\) and \(6x - 2\).

Step 2: Factor Out Common Factor in the First Pair

Look at the first pair, \(3x^3 - x^2\). Both terms have \(x^2\) as a common factor. Factoring \(x^2\) out gives us: \(x^2(3x - 1)\).

Step 3: Factor Out Common Factor in the Second Pair

Now, consider the second pair, \(6x - 2\). The common factor here is 2. Factoring out 2 gives: \(2(3x - 1)\).

Step 4: Combine the Groups

After factoring both pairs, we notice \(3x - 1\) is a common factor. We can write the expression as: \((x^2)(3x - 1) + 2(3x - 1)\).

Step 5: Factor Out the Common Binomial

Now, factor out \(3x - 1\) from the expression: \((3x - 1)(x^2 + 2)\).

Step 6: Write the Final Result

The expression \(3x^3 - x^2 + 6x - 2\) factors to: \((3x - 1)(x^2 + 2)\).

Key Concepts

Polynomial Expressions

A polynomial expression is a combination of terms where each term consists of a coefficient multiplied by variables raised to non-negative integer powers. In our exercise, the expression is \(3x^3 - x^2 + 6x - 2\). Each part of the expression, like \(3x^3\), is known as a term. The degree of the polynomial is determined by the term with the highest exponent. Here, \(3x^3\) makes it a third-degree polynomial.
Polynomial expressions can be added, subtracted, multiplied, or divided to form new polynomials. They're integral to algebra and higher mathematics, giving a structured way to express equations or mathematical relationships in a variety of fields.

Common Factor

A common factor in algebraic expressions is a term that divides all terms in the expression without leaving a remainder. In the provided exercise, common factors were used to simplify the expression by grouping terms. In the first pair, \(3x^3 - x^2\), the common factor is \(x^2\), as it appears in both terms. Factoring it out helps simplify the expression to \(x^2(3x - 1)\).
Similarly, in the second pair, \(6x - 2\), the common factor is 2. Extracting this factor results in \(2(3x - 1)\). Understanding how to identify and use common factors is crucial in simplifying and solving polynomial expressions efficiently.

Binomial Factoring

Binomial factoring is a method used to simplify polynomial expressions by recognizing and grouping terms into binomials that share a common factor. In our exercise, after factoring the common terms, we identified the terms \(3x - 1\) in both simplified groups: \(x^2(3x - 1)\) and \(2(3x - 1)\).
Seeing \(3x - 1\) as a common binomial allowed us to factor it out, simplifying the entire expression to \((3x - 1)(x^2 + 2)\). This step showcases how binomial factoring can reduce complex polynomial expressions to simpler, more manageable parts by identifying repeating binomial patterns.

Algebraic Expressions

Algebraic expressions consist of numbers, variables, and operations (such as addition and multiplication) but do not contain equality signs as equations do. The expression \(3x^3 - x^2 + 6x - 2\) is an example of an algebraic expression. These can be manipulated by operations like simplification, factoring, or expansion to solve algebraic problems.
Understanding algebraic expressions requires familiarity with the terminology like coefficients, variables, exponents, and terms, as well as how these interact in operations. Factoring, as demonstrated in the solution, shows how algebraic expressions can be broken down and simplified using techniques like recognizing common factors and grouping terms. These skills build the foundation for more advanced algebraic concepts and problem-solving strategies.