Problem 77 - Algebra A Combined Function

Question

Factor out the GCF from each polynomial. Then factor by grouping. $$ 12 x^{2} y-42 x^{2}-4 y+14 $$

Step-by-Step Solution

Answer

The GCF is 2; Factored expression: \(2(2y - 7)(3x^{2} - 1)\).

Step 1: Identify the GCF

Start by identifying the greatest common factor (GCF) of the coefficients. The coefficients of the terms are 12, -42, -4, and 14. The GCF of these numbers is 2.

Step 2: Factor out the GCF

Factor out the GCF, which is 2, from each term in the polynomial:\[2(6x^{2}y - 21x^{2} - 2y + 7)\]

Step 3: Group the Terms

Group the terms in pairs to make them easier to factor by grouping. Arrange and group them as follows:\[2((6x^{2}y - 21x^{2}) + (-2y + 7))\]

Step 4: Factor each Group

Factor out the common factors from each grouped pair. From \(6x^{2}y - 21x^{2}\), factor out \(3x^{2}\). From \(-2y + 7\), there are no common factors:\[2(3x^{2}(2y - 7) - 1(2y - 7))\]

Step 5: Factor out the Common Binomial

Observe that \(2y - 7\) is common in both grouped terms. Factor \(2y - 7\) out:\[2((2y - 7)(3x^{2} - 1))\]

Step 6: Write the Final Factored Expression

The final factored form of the given polynomial is:\[2(2y - 7)(3x^{2} - 1)\]

Key Concepts

Greatest Common Factor (GCF)

The Greatest Common Factor, or GCF, is the largest number that can divide all the terms of a polynomial without leaving a remainder. Finding the GCF is the first step in simplifying expressions and is particularly useful in polynomial factoring.

• To identify the GCF in a set of terms, focus on the coefficients and the variables separately.
  • For coefficients: analyze the numbers in each term, and determine the largest integer that divides them all.
  • For variables: find the lowest power of each variable common to all terms.
• Applying the GCF helps in factoring a polynomial by simplifying the terms, making further factorization steps much easier.

For example, in the polynomial given, the coefficients are 12, -42, -4, and 14. The largest number that evenly divides all these coefficients is 2. Thus, 2 is the GCF.

Factoring by Grouping

Factoring by grouping is a technique used to simplify polynomials with four or more terms. This method involves reorganizing and pairing terms to systematically reveal common factors.

• Rearrange terms to create groups that can be independently factored.
• Each group should contain terms that can be factored by a common factor other than 1.
• This technique works best when clumped pairs of terms share a common binomial factor.

In the polynomial we are factoring, first, we factor out the GCF, which is 2. This transformed our polynomial into \[ 2(6x^{2}y - 21x^{2} - 2y + 7) \]. After this, we identify two pairs \( (6x^{2}y - 21x^{2}) \) and \((-2y + 7) \) to perform factoring by grouping effectively.

Binomial Factorization

Binomial factorization is the process of expressing a polynomial as a product of binomials. It becomes evident once a common binomial is identified across grouped terms. It simplifies expressions and is an essential skill when aiming to simplify complex polynomials.

• Identify binomial patterns and similarities between different pairs of terms.
• Ensure the expression is structured such that the common binomial is evident in the groups' changes.
• Factor the polynomial by writing it as a product of the common binomial and the simplified terms.

In our scenario, after factoring each group, we focus on the expression \[ 2(3x^2(2y-7) - 1(2y-7)) \], where \((2y - 7) \) is a shared binomial factor. We then factor this out completely, resulting in the expression \[2((2y-7)(3x^2-1))\], simplifying the polynomial into a neat binomial factorization.