Problem 71 - Algebra A Combined Function

Question

Factor each four-term polynomial by grouping. If this is not possible, write "not factorable by grouping." $$ 3 x-3+x^{3}-4 x^{2} $$

Step-by-Step Solution

Answer

Not factorable by grouping.

Step 1: Group the Terms

Rearrange the polynomial terms so that similar components are grouped together. The polynomial is given as: \[ x^3 - 4x^2 + 3x - 3 \] Group the terms:\[ (x^3 - 4x^2) + (3x - 3) \]

Step 2: Factor Each Group

Factor out the greatest common factor from each group: \[ x^2(x - 4) + 3(x - 1) \]

Step 3: Check Common Binomial

Observe if there is a common binomial factor in each group. In this case, the binomials \(x - 4\) and \(x - 1\) are different, so there is no common factor.

Step 4: Conclusion

Since there is no common binomial factor in the terms factored by grouping, the polynomial \(3x - 3 + x^3 - 4x^2\) is not factorable by grouping.

Key Concepts

Polynomial Expressions

A polynomial expression is a fundamental concept in algebra that involves sums of various terms. Each term consists of a variable (or variables) raised to a power and multiplied by a coefficient. The expression we are dealing with here is \(x^3 - 4x^2 + 3x - 3\). In this polynomial:

• \(x^3\) is the cubic term, which has the highest degree of 3.
• \(-4x^2\) is the quadratic term.
• \(3x\) is the linear term.
• \(-3\) is the constant term.

Each of these components is part of the polynomial's structure, contributing to its overall expression. Key to working with polynomials is understanding how to rearrange and manipulate these terms, especially when trying to simplify or factor expressions.

Grouping Method

The grouping method is a popular technique for factoring polynomials, especially when dealing with polynomials with four terms. The basic idea is to group terms into pairs that have common factors, then factor these pairs separately. Let's break it down:

• First, rearrange the terms if necessary to identify pairs that can be grouped together effectively. In our example, this yields the grouping \((x^3 - 4x^2) + (3x - 3)\).
• Second, look for and extract the greatest common factor (GCF) from each group.

This approach simplifies the polynomial into smaller, manageable parts. However, if the grouped terms do not share a common factor, the method may not result in a fully factored expression, as seen in our solution where no common binomial factor emerged.

Greatest Common Factor

The greatest common factor (GCF) is a key concept when factoring polynomials, especially in algebraic expressions. It refers to the largest factor that divides all terms within a particular expression or group. In the context of the grouping method:

• Identify the common factor within each pairing. For \(x^3 - 4x^2\), the GCF is \(x^2\), and for \(3x - 3\), it is 3.
• Factor these GCFs out of their respective groups, simplifying each grouping: \(x^2(x - 4) + 3(x - 1)\).

Factoring out the GCF is a crucial step in simplifying an expression and often makes visible the underlying structure or patterns that can lead to further factorization. It's an invaluable skill in both basic and advanced algebra.

Binomials

A binomial is a polynomial expression that consists of exactly two terms, for example, \(x + 1\) or \(2x - 3\). When using the grouping method, identifying common binomial factors is crucial for complete factorization. In our exercise, after grouping and factoring, we obtained two binomials:

• \(x - 4\)
• \(x - 1\)

A successful grouping results in a common binomial factor, which did not occur here as the two binomials differ. When no such common binomial is present, as noted, the expression cannot be factored further by this method. Recognizing binomials is important, as they often play a key role in simplifying and solving algebraic expressions.