Problem 49 - Algebra A Combined Function

Question

Factor each trinomial completely. Some of these trinomials contain a greatest common factor (other than 1). Don't forget to factor out the GCF first. $$ 4 x^{2} y+4 x y-12 y $$

Step-by-Step Solution

Answer

The factored form is \(4y(x + 3)(x - 1)\).

Step 1: Identify the Greatest Common Factor (GCF)

Look at the coefficients and variables in each term of the trinomial: \(4x^2y\), \(4xy\), and \(-12y\). The coefficients 4, 4, and -12 have a common factor of 4. Each term also includes the variable \(y\). So, the GCF of the terms is \(4y\).

Step 2: Factor Out the GCF

Factor the GCF \(4y\) out of each term in the trinomial to simplify it. This is done as follows: \[4x^2y + 4xy - 12y = 4y(x^2 + x - 3).\]

Step 3: Factor the Remaining Trinomial

Look at the trinomial \(x^2 + x - 3\). We need to find two numbers that multiply to \(-3\) (the constant term) and add up to \(1\) (the coefficient of the \(x\) term). These numbers are \(3\) and \(-1\). Factor the trinomial as follows: \[(x^2 + x - 3) = (x + 3)(x - 1).\]

Step 4: Combine the Factors

After factoring, combine the terms: the factored expression is \[4y(x + 3)(x - 1).\] Ensure that all parts are correctly included: the \(4y\) is the GCF and \((x + 3)(x - 1)\) is the factored form of the trinomial.

Key Concepts

Greatest Common Factor (GCF)

Understanding the Greatest Common Factor (GCF) is crucial in simplifying algebraic expressions. The GCF is the largest factor shared by all terms in an expression. It includes both numerical coefficients and variables. For instance, in the trinomial given: - Terms are: \(4x^2y\), \(4xy\), and \(-12y\).- Coefficients are 4, 4, and -12. Their GCF is 4. - Each term also contains at least one \(y\). Therefore, the GCF of the entire expression is \(4y\).
To factor an expression quickly, identify the biggest number and the highest power of variables common in all terms. Factoring the GCF first simplifies the resulting expression, making further factoring easier.

Algebraic Expressions

Algebraic expressions are a mixture of numbers, variables, and operators (like addition and multiplication) combined together. They can be as simple as a single variable or number, or as complex as polynomials.
Each expression consists of terms, which may include:- Coefficients: Numerical part that multiplies the variables.- Variables: Letters representing unknown values.- Constants: Fixed numerical values.
In our exercise, \(4x^2y + 4xy - 12y\) is the expression we're working with. Here, the three terms are connected through addition/subtraction and involve variables like \(x\) and \(y\). Understanding each component's role is key to manipulating and simplifying these expressions correctly.

Polynomial Factoring

Factoring polynomials involves breaking down a complex polynomial into simpler components, or factors, that when multiplied together give the original polynomial. This simplification can make it easier to solve equations or understand relationships between variables.
In our example, after factoring out the GCF, you are left with \(x^2 + x - 3\). This trinomial can be factored further. You look for two numbers that when multiplied equal the constant term, \(-3\), and when added equal the coefficient of the linear term, \(1\). These numbers are \(3\) and \(-1\), so the trinomial factors into \((x + 3)(x - 1)\).
Factoring is like decomposing a puzzle into pieces that fit together in a certain way. Doing so requires recognizing patterns and understanding the structure of polynomials.