Problem 88 - Elementary Algebra

Question

Factor by grouping. $$ 2 x 4-x 3-6 x+3 $$

Step-by-Step Solution

Answer

The expression is factored as \((2x - 1)(x^3 - 3)\).

Step 1: Rewrite the Expression

Start by rewriting the given expression in a more readable format: \(2x^4 - x^3 - 6x + 3\). The expression needs is already formatted to be factored using grouping.

Step 2: Group the Terms

Split the expression into two groups: \((2x^4 - x^3)\) and \((-6x + 3)\). This will allow us to factor each group separately.

Step 3: Factor Each Group Separately

For the first group \((2x^4 - x^3)\), factor out \(x^3\): \(x^3(2x - 1)\). For the second group \((-6x + 3)\), factor out \(-3\): \(-3(2x - 1)\).

Step 4: Combine the Factors

Both factored groups \(x^3(2x - 1)\) and \(-3(2x - 1)\) have a common factor \((2x - 1)\). Factor \((2x - 1)\) out: \((2x - 1)(x^3 - 3)\).

Step 5: Write the Final Factored Form

The expression \(2x^4 - x^3 - 6x + 3\) can be factored as \((2x - 1)(x^3 - 3)\).

Key Concepts

Factor by Grouping

Factoring by grouping is a powerful algebraic technique used to simplify polynomial expressions by breaking them down into simpler, easier-to-manage parts.
Instead of dealing with a complex expression in its entirety, we strategically split it into groups, usually of two terms each. This method is particularly useful when a polynomial doesn't have a clear monomial factor common to all terms.
The steps are straightforward:

• Identify pairs of terms that can form groups, typically, this will include rearranging the terms if needed.
• Factor each group separately, as if solving a smaller and simpler problem.
• Look for a common factor across the groups, which can usually be factored out.
• Combine the factored expressions for the final result.

By breaking the expression into smaller pieces, factoring by grouping makes complex problems manageable. Remember, practice makes perfect with this technique!

Polynomial Expressions

A polynomial expression comprises variables and coefficients, constructed using operations of addition, subtraction, multiplication, and non-negative integer exponents.
Such expressions can be intimidating owing to their sometimes complicated appearance, but they are fundamental to algebra and calculus.
Polynomials are categorized based on the number of terms:

• Monomial: An expression with only one term (e.g., \(3x^2\)).
• Binomial: Comprised of two terms (e.g., \(4x^2 + 2x\)).
• Trinomial: Contains three terms (e.g., \(x^2 - 5x + 6\)).

Understanding the structure of polynomial expressions can help in identifying suitable algebraic techniques for factoring them. Whether a polynomial has three terms or more, each challenge can be approached with confidence by applying the right method, such as grouping, to simplify it.

Algebraic Factoring Techniques

Algebraic factoring techniques are essential tools for simplifying and solving polynomial expressions. These techniques involve breaking down complex equations into their simpler, constituent factors.
One of the most common techniques is "factoring by grouping," but several other methods are also widely used:

• Factoring out the Greatest Common Factor (GCF): Identifying the largest factor shared by all terms in a polynomial is often the first step in simplifying an expression.
• Difference of Squares: This approach is used when dealing with polynomials in the form of \(a^2 - b^2\), which can be factored into \((a + b)(a - b)\).
• Trinomial factoring: Specific to expressions with three terms, it often involves finding two numbers that multiply to the constant term and add up to the coefficient of the middle term.

By mastering various algebraic factoring techniques, students can gain a robust toolkit for tackling a wide range of mathematical problems. Each technique unlocks new avenues for simplifying polynomials and solving equations effectively.