Problem 28 - Elementary and Intermediate Algebra

Question

Factor each polynomial. $$ 14 s^{3}+15 t^{6} $$

Step-by-Step Solution

Answer

The polynomial cannot be factored further; it remains as \(14s^3 + 15t^6\).

Step 1: Identify the Greatest Common Factor (GCF)

First, examine the polynomial \(14s^3 + 15t^6\) to identify any common factors. The coefficients 14 and 15 have no common factors other than 1, and the terms involve different variables \(s\) and \(t\). Thus, there is no non-trivial GCF apart from 1.

Step 2: Check for Factorization by Other Methods

Since the terms involve different variables and no binomial form like difference of squares or sum of cubes is applicable here, we can't directly factor this polynomial further. This polynomial is already in a factored state as much as possible given the provided terms.

Key Concepts

Greatest Common Factor

The greatest common factor (GCF) is the largest factor that divides two or more numbers. In the context of polynomials, it refers to the largest polynomial that divides each term of a polynomial expression. Finding the GCF is often the first step in factoring polynomials, as it simplifies the expression and can make the next steps easier. To find the GCF, examine the coefficients of each term and the variables included. For the polynomial expression \(14s^3 + 15t^6\), consider:

• Coefficients: 14 and 15
• Variables: \(s\) and \(t\)

The numbers 14 and 15 share no common factors other than 1. The variables \(s\) and \(t\) are different, which means no common variable factor. Thus, the GCF for this polynomial is simply 1. This means there is no need for additional factoring based on the GCF alone. Recognizing when the GCF is 1 is crucial as it guides you to consider other factoring methods.

Polynomial Expressions

Polynomial expressions are algebraic expressions made up of terms, where each term consists of a constant coefficient, variables, and their exponents. Understanding polynomials is essential to mastering algebraic factoring. A polynomial looks like \(a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0\), where each \(a\) represents a coefficient and the exponents are non-negative integers. Key features of polynomial expressions:

• Terms: Each part of the polynomial separated by plus or minus signs.
• Degree: The highest exponent value of the variable in a term.
• Coefficient: A number multiplying the variable in a term.

For \(14s^3 + 15t^6\), there are two terms: \(14s^3\) and \(15t^6\). The polynomial is of a simple form with each term involving a different variable and neither sharing any factors. This uniqueness in the terms implies limited factoring possibilities, as seen in this example where no further simplification or factoring apart from GCF can be applied.

Algebraic Factoring Methods

Algebraic factoring methods are strategies used to express a polynomial as a product of simpler polynomials. While factoring, the goal is to simplify the polynomial or to write it in a product form that reveals roots or solutions. Different methods apply depending on the type of polynomial we encounter.Some common algebraic factoring methods include:

• Factoring out the GCF
• Difference of squares
• Sum or difference of cubes
• Using the quadratic formula

In our exercise, the polynomial \(14s^3 + 15t^6\) doesn't conform to special factoring formulas such as difference of squares or sum of cubes, primarily because each term consists of different variables \(s\) and \(t\). This means that after confirming the GCF of 1, no further algebraic factoring methods are applicable. Understanding this limits unnecessary steps and directs focus on specific factoring strategies only when applicable. Knowing when you have fully factored a polynomial helps in further studies like solving equations using these expressions.