Problem 17 - Algebra 2

Question

Factor completely. If the polynomial is not factorable, write prime. $$ 3 a^{2} b x+15 c x^{2} y+25 a d^{3} y $$

Step-by-Step Solution

Answer

The polynomial is prime and cannot be factored.

Step 1: Identify the Greatest Common Factor (GCF)

Examine the coefficients and the variable parts of each term: \(3a^2bx\), \(15cx^2y\), and \(25ad^3y\). The GCF of the coefficients (3, 15, and 25) is 1. For the variable part, no variable is common in all terms. Hence, GCF in terms of variable does not exist.

Step 2: Factor by Grouping Strategy Attempt

Attempt to arrange or group terms such that a common factor might emerge, but in this polynomial, rearranging terms like \((3a^2bx + 15cx^2y) + 25ad^3y\) or other combinations doesn't result in a common factor in either sets of terms, leaving no further factoring option available.

Step 3: Assess for Special Factoring Formulas

Check if the polynomial fits any known special factoring formulas such as difference of squares, perfect square trinomials, or sum/difference of cubes. None of these formulas fit the structure of this polynomial.

Step 4: Conclusion – Prime Polynomial

Since there is no factor common to all terms or any possibility for grouping or special formulas, the polynomial cannot be factored further using integer coefficients and exponents. The final determination is that the polynomial is prime.

Key Concepts

Greatest Common Factor (GCF)

When factoring polynomials, the first step is to search for the Greatest Common Factor (GCF). The GCF is the highest number or expression that divides exactly into each term of the polynomial. To find the GCF, you need to consider both numerical coefficients and variables:

• Numerical Coefficients: For this polynomial, the coefficients are 3, 15, and 25. The largest number that divides all of them is 1. Hence, there is no common numerical factor larger than 1.
• Variables: Check for common variables in all terms. Analyze terms such as \(3a^2bx\), \(15cx^2y\), and \(25ad^3y\). Since no variable appears in every term, a GCF from variables does not exist.

The absence of a GCF means we cannot initially simplify or reduce the expressions by common factors. Understanding this basic principle helps in assessing what possible methods, if any, could further simplify the polynomial.

Prime Polynomial

A polynomial is considered a "Prime Polynomial" when it cannot be factored into the product of two non-trivial polynomials having integer coefficients. It's similar to prime numbers but with polynomials. In the given exercise:

• The attempt to find a common factor across terms yielded that no such common entity exists beyond '1' and variable commonality was lacking.
• The option to factor by grouping, also failed as no useful grouping emerged that presented common factors across either group.

Upon failing both common factor extraction and the grouping strategy, we label the polynomial as prime. Recognizing prime nature is crucial as it saves time from futile attempts to factor further.

Special Factoring Formulas

Special factoring formulas provide shortcuts to factor certain types of polynomials quickly. Common methods include:

• Difference of Squares
• Perfect Square Trinomials
• Sum/Difference of Cubes

These formulas offer structured techniques to factor specific patterns effectively. For the polynomial in question, no standard formula could be applied:

• The polynomial does not resemble a difference of squares since it contains three terms and involves variable multiplicity and mix.
• It fails to match the perfect square trinomial pattern as term structure diverges from required criteria.
• Lastly, presence of varied terms rules out sum or difference of cubes.

Overall, understanding these formulas ensures you quickly identify and apply them when necessary, though some polynomials, like this one, won't comply with any special forms given.