Question

Factor: \(3 x^{3} y-2 x^{2} y^{2}+18 x-12 y\)

Short answer

x[y(3x² - 2xy) + 6(3 - 2y)]

Step-by-step solution

Identify the greatest common factor (GCF)

The terms of the expression are: 1. 3x³y 2. -2x²y² 3. 18x 4. -12yFirst, find the GCF of the numerical coefficients: The GCF of 3, -2, 18, and -12 is 1.Now, find the GCF of the variables in the terms: The GCF for the variables is 'x' (since it is common in terms involving 'x').So, the overall GCF of the expression is x.

Factor out the GCF

Factor x out of each term in the expression: Starting expression: 3x³y - 2x²y² + 18x - 12yFactoring out x from each term: x(3x²y - 2xy² + 18 - 12y)

Factor by grouping

Group terms in pairs to simplify further: x[(3x²y - 2xy²) + (18 - 12y)]Factor out common terms within each group: x[y(3x² - 2xy) + 6(3 - 2y)]

Final result

Combine all the factors to get the final factorization: x[y(3x² - 2xy) + 6(3 - 2y)]

Key concepts

greatest common factor

The greatest common factor (GCF) is the largest factor that divides all the given terms without leaving a remainder. To start, focus on the numerical coefficients of each term in the polynomial. For example, in the expression \(3x^{3}y - 2x^{2}y^{2} + 18x - 12y\), the numerical coefficients are 3, -2, 18, and -12. The GCF of these numbers is 1.
Next, look at the variables in each term. In the example, 'x' is common in terms involving 'x', which makes it part of the GCF for the variables.
Therefore, the overall GCF for the entire polynomial is x.

factor by grouping

Factoring by grouping is a useful method, especially when dealing with longer polynomials. The goal is to form groups of terms within the polynomial that can be factored further.
To illustrate, let's work with the expression from our exercise after extracting the GCF: \(x(3x^{2}y - 2xy^{2} + 18 - 12y)\).
Group the terms in pairs: \(x[(3x^{2}y - 2xy^{2}) + (18 - 12y)]\).
Within each group, we can now focus on factoring out the common terms. For the first group \((3x^{2}y - 2xy^{2})\), 'y' is common, and for the second group \((18 - 12y)\), '6' is common.
Factoring out these common terms, we get: \(x[y(3x^{2} - 2xy) + 6(3 - 2y)]\).

algebraic expressions

An algebraic expression is a mathematical phrase involving numbers, variables and operation symbols. Expressions can range from simple (like \(5x + 3\)) to complex combinations (such as the polynomial we're working with: \(3x^{3}y - 2x^{2}y^{2} + 18x - 12y\)).
Understanding how to break down and manipulate these expressions is key to solving algebraic problems. Simplifying, factoring, and combining like terms are standard procedures used in dealing with algebraic expressions.
Our initial step is identifying the GCF before moving on to factor by grouping, which simplifies our complex polynomials.

polynomial decomposition

Polynomial decomposition refers to breaking down a polynomial into simpler factors that, when multiplied together, give back the original polynomial. This makes it easier to solve, analyze or simplify equations.
In our example, after factoring out the GCF \(x\), then applying the grouping method, we simplified \(3x^{3}y - 2x^{2}y^{2} + 18x - 12y\) to \(x[y(3x^{2} - 2xy) + 6(3 - 2y)]\).
By breaking down the polynomial in steps, we make complex expressions more manageable, allowing us to further solve or simplify them efficiently.

• So first, identify the GCF.
• Next, group the terms.
• Then, factor out the common terms within each group.

Following these steps makes polynomial decomposition a systematic and straightforward process.