Question

$$4 x^{2} y+c x y^{2}+3 x y^{3}$$

Short answer

The factored form of the polynomial is \( xy(4x + cy + 3y^2) \).

Step-by-step solution

Identify Common Factors

Inspect the terms of the polynomial to identify any common factors. In this case, all terms have an 'x' and a 'y' in them.

Factor out Common Variables

Since each term of the polynomial contains at least one 'x' and one 'y', factor out the common variables which is 'xy' from each term.

Rewrite the Polynomial

After factoring out the common 'xy', rewrite the polynomial as the product of 'xy' and the simplified terms inside the parenthesis. The simplified polynomial is: \( xy(4x + cy + 3y^2) \)

Key concepts

Understanding Common Factors in Polynomial Factoring

Factoring polynomials is a fundamental skill in algebra that simplifies expressions and solves equations. The process often begins with identifying common factors, which are numbers or variables that are present in all terms of the polynomial.

For example, consider the polynomial \(4x^2y+ cxy^2 + 3xy^3\). Here, each term has an 'x' and a 'y' in them, making 'xy' a common factor. By recognizing this, you can pull 'xy' out front, effectively reducing the complexity of the polynomial. It's akin to untying a knot - once the common thread is pulled out, what remains is more manageable and easier to work with.

Think of common factors as the glue holding terms together; by isolating them, you can see the individual components more clearly. This step is crucial because it sets the stage for further simplification and allows for operations such as addition, subtraction, or even more advanced factoring techniques.

Variables in Polynomials and Their Roles

Polynomials are algebraic expressions that consist of variables and coefficients, combined using the operations of addition, subtraction, and multiplication. Variables in polynomials are represented by letters and may be accompanied by exponents to denote repeated multiplication.

When dealing with variables in polynomials, it's important to understand the power of each variable. For instance, in the term \(x^2y\), 'x' is raised to the second power while 'y' is not, indicating that there are two 'x' factors but only one 'y'. In the expression \(4x^2y+ cxy^2 + 3xy^3\), the variables 'x' and 'y' are found with varying exponents, which affects how the polynomial can be factored.

Identifying the Roles

Variables act as placeholders to represent unknown values and are essential for defining the shape and properties of the graph of a polynomial function. Recognizing the highest power of a variable gives insight into the potential complexity and the number of roots or solutions that polynomial can have. Hence, a firm grasp on the function of variables is pivotal in mastering polynomial manipulation.

Simplifying Polynomials for Easier Understanding

The goal of simplifying polynomials is to transform complex algebraic expressions into their simplest form, making them more comprehensible and easier to evaluate. Simplification involves several techniques, including factoring, combining like terms, and applying the distributive property.

A simplified polynomial is more than just a shorter expression; it's a clearer representation of the problem at hand, allowing for straightforward analysis and solution. In our example \(4x^2y+ cxy^2 + 3xy^3\), after factoring out the common 'xy', we rewrote the polynomial as \(xy(4x + cy + 3y^2)\). This new expression is visually less complicated and reveals the polynomial's inherent structure.

Benefits of Simplification

Simplifying polynomials makes it easier for anyone working with the expression to perform operations, find roots, and even graph the function. It can be especially helpful for students who are just starting to learn about polynomial functions and need clear, concise steps to follow. Overall, a simplified polynomial paves the way for a better understanding and greater success in algebra.