Problem 54 - Elementary and Intermediate Algebra

Question

Factor. $$ 12 x^{3} z+12 x y^{2} z-8 x^{2} y z-8 y^{3} z $$

Step-by-Step Solution

Answer

The factored expression is \( 4xz((3x - 2y)(x + y^2)) \).

Step 1: Identify the greatest common factor

First, identify the greatest common factor among all the terms. All the terms in the expression have a common factor of \( 4xz \). This will simplify the factoring process.

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

Extract \( 4xz \) from each term of the expression:\[12x^3z + 12xy^2z - 8x^2yz - 8y^3z = 4xz(3x^2 + 3y^2 - 2xy - 2y^3)\]This leaves us with a secondary polynomial that needs to be factored further.

Step 3: Group terms to simplify factoring

Next, rearrange and group the terms of the inner polynomial expression to make them easier to factor. Observe that they can be grouped into pairs:\[4xz((3x^2 - 2xy) + (3y^2 - 2y^3))\]

Step 4: Factor each group separately

From the grouped terms, factor each pair individually:\[3x^2 - 2xy = x(3x - 2y)\]\[3y^2 - 2y^3 = y^2(3 - 2y)\]Thus, our expression becomes:\[4xz(x(3x - 2y) + y^2(3 - 2y))\]

Step 5: Utilize the distributive property

Recognize that the terms can be added using the distributive property inside the parenthesis:\[4xz((3x - 2y)(x + y^2))\]

Step 6: Final Step: Verify the factorization

Check the solution by expanding the factors to ensure it results in the original expression:Expanding the factors within the parenthesis:\[(3x - 2y)(x + y^2) = 3x^2 + 3xy^2 - 2xy - 2y^3\]Each term matches the secondary polynomial found by factoring out in Step 2, confirming the factorization is correct.

Key Concepts

Greatest Common Factor

In algebra, the greatest common factor (GCF) is a key element in simplifying expressions and solving polynomial equations. It is the highest number or expression that can divide all terms of a given polynomial without leaving a remainder. Identifying the GCF can simplify the factorization process significantly.

To find the GCF in polynomial expressions:

• Look for the common numerical factor of all coefficients in the terms.
• Identify any common variables, and choose the lowest power of these variables that is present in each term.

In our example, the terms are 12x³z, 12xy²z, -8x²yz, and -8y³z. Observing these, all terms share 4 as a common numerical factor, "x" as a common variable to the first power, and "z" as it's a common factor in each term. Thus, the GCF for the expression is 4xz, and extracting this simplifies the expression immediately. Finding the GCF is the first and crucial step for any factorization task.

Polynomial Grouping

Polynomial grouping is a valuable technique that allows for breaking down complex polynomials into smaller, more manageable groups. This is especially useful when dealing with higher degree polynomials that do not have a straightforward GCF.

Here is how polynomial grouping works:

• Look for terms in the polynomial that can be paired based on similar characteristics.
• Group them in such a way that each group can be factored easily.
• This often involves rearranging terms to find suitable groupings.

In the given example, after factoring out the GCF 4xz, the polynomial is 3x² + 3y² - 2xy - 2y³. These terms can be grouped into (3x² - 2xy) and (3y² - 2y³). By creating these groups, each one can be factored individually, simplifying the polynomial into a product of smaller and simpler expressions.

Distributive Property

The distributive property is a fundamental concept in algebra that allows you to simplify expressions by distributing multiplication over addition or subtraction. It is expressed as \( a(b + c) = ab + ac \). This property is crucial when combining terms to aid in factorization.

In factorization, once you've grouped terms and factored them individually, the distributive property lets you combine them effectively:

• Recognize common factors within the groups that can be combined under one expression, which is often enclosed within parentheses.

In our example, the expression after grouping and factoring individually becomes 4xz(x(3x - 2y) + y²(3 - 2y)). Observing the factors and applying the distributive property, this can be rewritten as 4xz((3x - 2y)(x + y²)).

This step ensures the expression is fully simplified and correctly factored. It also helps verify the solution by checking back with the initial polynomial to confirm all terms are accounted for.