Problem 76 - Elementary Algebra

Question

Factor completely. Identify any prime polynomials. $$ 12 x^{2} z+4 x^{2}-3 x y z-x y $$

Step-by-Step Solution

Answer

The completely factored form is \((3z + 1)(4x^2 - xy)\). Neither factor is prime.

Step 1: Group the terms

Group the polynomial into pairs to facilitate factoring by grouping.\[ (12x^2z + 4x^2) + (-3xyz - xy) \]

Step 2: Factor out the greatest common factor (GCF) from each pair

Factor out the GCF from each group of terms. For the first group, the GCF is \(4x^2\), and for the second group, the GCF is \(-xy\).\[ 4x^2(3z + 1) - xy(3z + 1) \]

Step 3: Factor out the common binomial

Notice that \( (3z + 1) \) is common in both terms. Factor \( (3z + 1) \) out.\[ (3z + 1)(4x^2 - xy) \]

Step 4: Identify if any more factors can be extracted

Check if \( (4x^2 - xy) \) can be factored further. Since it cannot be factored further, the factorization process is complete.

Key Concepts

Grouping Terms

To factor a polynomial, it's often helpful to group terms together. This method works best when there are at least four terms to work with. We start by splitting the polynomial into groups. For example, in the polynomial those groups look like this: Grouping like this makes it clearer to see any common factors within each group. The goal is to find pairs of terms that have a common factor. This setup will help in the next step of factoring.

Greatest Common Factor (GCF)

The greatest common factor (GCF) is the largest factor shared between terms in a polynomial expression. In our grouped polynomial: we look for the GCF in each group. For the first group, is a multiple of , and is a multiple of . So, is the GCF for the first group. For the second group, is a multiple of y, and x y is also a multiple of y. Hence, y is the GCF for the second group. Using the GCF, we can simplify each part of the polynomial: simplifies to and simplifies to Now our polynomial looks like this:
This step makes it easier to factor further or to identify patterns within the polynomial.

Prime Polynomial

A prime polynomial is a polynomial that cannot be factored further other than by multiplying it by 1 or -1. In our example, after extracting the common binomial (3z + 1), we observed the term: We checked if can be factored further. Since it cannot be factored further using integers and rational numbers, we recognize it as a prime polynomial. This identification is crucial because sometimes recognizing when a polynomial is prime can save you from needless efforts to factor further.

Binomial

A binomial is a polynomial with exactly two terms. When factoring polynomials, sometimes we end up with binomials. In our example, we recognized: (3z + 1) as a common factor, which is a binomial. Extracting this common binomial simplified our expression to: Remember that binomials are often seen in factoring quadratic expressions as well. If you remember key patterns, like difference of squares ( a^2 - b^2) and perfect square trinomials ( a^2 + 2ab + b^2), it can make factoring much easier and faster.