Question

Simplify: \(x^{4} y^{2}+x^{3} y^{5}+x y^{6}+2 x^{3} y^{5}\) (A) \(x y\left(x^{4}+3 x^{2} y^{3}+y^{6}\right)\) (B) \(y^{2}\left(x^{4}+4 x^{3} y^{4}+y^{3}\right)\) (C) \(x y^{2}\left(x^{3}+3 x^{2} y^{3}+y^{4}\right)\) (D) \(x\left(x^{3} y^{2}+x^{2} y^{5}+y^{6}+2 x^{4} y^{4}\right)\)

Short answer

The simplified expression is \(x y^2(x^3 + 3x^2 y^3 + y^4)\), which matches option (C).

Step-by-step solution

Identify Common Factors

Let's first analyze each term in the polynomial expression: - The terms are \(x^4 y^2\), \(x^3 y^5\), \(x y^6\), and \(2 x^3 y^5\).- We will determine the greatest common factor among them. Notice that all terms have at least one 'x' and two 'y's as factors.

Factor out the Greatest Common Factor

The greatest common factor for the terms is \(x y^2\). Factoring this out from each term gives us:- From \(x^4 y^2\), we factor out \(x y^2\) and get \(x^3\).- From \(x^3 y^5\), we factor out \(x y^2\) and get \(x^2 y^3\).- From \(x y^6\), we factor out \(x y^2\) and get \(y^4\).- From \(2 x^3 y^5\), we factor out \(x y^2\) and get \(2x^2 y^3\).

Write the Factored Expression

After factoring out \(x y^2\), we rewrite the expression as: \[ x y^2 (x^3 + 3x^2 y^3 + y^4) \].

Compare with Given Choices

Now, we compare the factored expression \(x y^2 (x^3 + 3x^2 y^3 + y^4)\) with the provided options: - Option (A): \(x y(x^4 + 3x^2 y^3 + y^6)\)- Option (B): \(y^2(x^4 + 4x^3 y^4 + y^3)\)- Option (C): \(x y^2(x^3 + 3x^2 y^3 + y^4)\)- Option (D): \(x(x^3 y^2 + x^2 y^5 + y^6 + 2x^4 y^4)\)- The correct match is Option (C).

Key concepts

Polynomial Factorization

Polynomial factorization is a technique used in algebra to express a polynomial as the product of its factors. These factors can be numbers, variables, and other polynomials. Factoring is a useful tool because it simplifies expressions, making them easier to understand or solve.

In our exercise, the expression is given as a polynomial with four terms:

• \(x^4y^2\)
• \(x^3y^5\)
• \(xy^6\)
• \(2x^3y^5\)

The goal of polynomial factorization is to write this expression in a simpler form. To do so, we look for terms that share common numerical or variable factors. By identifying and extracting these common factors, the expression is significantly reduced in complexity. Factoring can be incredibly valuable, as it often reveals the underlying structure or useful identities within the polynomial that would otherwise be difficult to see.

Greatest Common Factor

The Greatest Common Factor (GCF) is the largest factor that divides two or more numbers, or in our case, terms of a polynomial. Finding the GCF is an essential step in simplifying expressions through factorization.

Let's look at the given polynomial expression:

• Term 1: \(x^4y^2\)
• Term 2: \(x^3y^5\)
• Term 3: \(xy^6\)
• Term 4: \(2x^3y^5\)

To find the GCF, we examine each term to identify the lowest powers of common variables. Each term has at least one \(x\) and two \(y\)'s. Therefore, the GCF of this polynomial is \(xy^2\). By factoring \(xy^2\) out from each term, we simplify the polynomial into its essential components. Finding the GCF is a fundamental skill in algebra as it aids in the process of factoring, an important method for manipulating and simplifying algebraic expressions.

Expression Simplification

Expression simplification involves reducing an algebraic expression to its simplest form without changing its value. Through simplification, we aim to make the expression easier to work with by removing any unnecessary complexity. In our example, after finding the GCF \(xy^2\), we simplify each term as follows:

• From \(x^4y^2\), factoring out \(xy^2\) results in \(x^3\).
• From \(x^3y^5\), factoring out \(xy^2\) results in \(x^2y^3\).
• From \(xy^6\), factoring out \(xy^2\) results in \(y^4\).
• From \(2x^3y^5\), factoring out \(xy^2\) results in \(2x^2y^3\).

By pulling out the common factor and rewriting the expression, we significantly reduce its complexity. The simplified form \(xy^2(x^3 + 3x^2y^3 + y^4)\) is both elegant and far easier to assess or solve further problems with. Simplification is often used to make solving equations more straightforward or to prepare an expression for further operations or analysis.