Problem 76 - Algebra 2

Question

Factor completely. If the polynomial is not factorable, write prime. $$ 13 x y z+3 x^{2} z+4 k $$

Step-by-Step Solution

Answer

The polynomial is prime.

Step 1: Identify the Terms and Variables

The polynomial presented is \(13xyz + 3x^2z + 4k\). It contains three terms: \(13xyz\), \(3x^2z\), and \(4k\). The variable parts are \(xyz\), \(x^2z\), and \(k\).

Step 2: Check for Common Factors

Assess if there is a common factor among all three terms. The first term has factors \(13, x, y, z\), the second has \(3, x, x, z\), and the third has \(4, k\). There are no common factors shared by all the terms.

Step 3: Analyze for Pairwise Factoring

Examine pairs of terms to see if any two terms share a common factor. For \(13xyz\) and \(3x^2z\), the common factor is \(xz\). Factor \(xz\) out: \(xz(13y + 3x)\). However, \(4k\) is not factorable with this group.

Step 4: Conclude on Factorability

Since there is no common factor for all terms or a factorable pair including all three terms, the polynomial cannot be factored further using basic methods.

Key Concepts

Common Factors

In polynomial factoring, identifying common factors is a crucial first step. A common factor is a number or variable shared by each term in a polynomial. Recognizing these can simplify expressions by grouping shared elements.

In the exercise given, the polynomial is \(13xyz + 3x^2z + 4k\). Upon examining each term's factors:

• The first term \(13xyz\) includes the factors \(13, x, y, z\).
• The second term \(3x^2z\) consists of \(3, x, x, z\).
• The third term \(4k\) is simply \(4, k\).

Checking across these terms, no factor is shared universally.

This lack of a common factor means we can't simply "pull out" a term from every part of the polynomial, which often is the easiest way to reduce its complexity.

Terms and Variables

Understanding the distinction between terms and variables is key for polynomial factoring. A term is a part of an expression that is separated by a plus or minus sign, while a variable is a symbol representing numbers in a term.

In our polynomial \(13xyz + 3x^2z + 4k\), we have three distinct terms:

• \(13xyz\)
• \(3x^2z\)
• \(4k\)

Each term consists of its coefficients and variables. The variables \(x, y, z,\) and \(k\) appear within these terms. Each variable holds a specific role and potential value in calculations.

Recognizing these separate elements is essential. Being able to identify what variables are at play helps us determine the relationships within each term and their role in possible factoring opportunities.

Polynomial Factorability

Polynomial factorability checks whether a polynomial can be broken down into simpler polynomials multiplied together. Not all polynomials have this property, making this step critical.

For the polynomial \(13xyz + 3x^2z + 4k\), after checking for common factors without success, the next step is pairwise factoring. Here, we look for shared factors among pairs of terms:

• \(13xyz\) and \(3x^2z\) share an \(xz\). Factoring this gives \(xz(13y + 3x)\).
• The term \(4k\) doesn't fit into this pair, lacking connections enough for factorization.

Since no common factor exists for all terms and no satisfactory grouping includes each term, we declare the polynomial prime. This means it’s not further factorable via simple methods, and stands alone as irreducible.