Problem 32 - Elementary Algebra

Question

For the following problems, factor the polynomials. $$ 2 y^{2}+6 y+4 x y $$

Step-by-Step Solution

Answer

Answer: $$2(y^2 + y(3 + 2x))$$.

Step 1: Identifying Common Factors

We have the polynomial $$2y^2 + 6y + 4xy$$. Let's identify any common factors among the terms. In this case, we see that all three terms have a common factor of 2.

Step 2: Factoring Out Common Factors

Now that we have identified the common factor, we can factor it out: $$2(y^2 + 3y + 2xy)$$.

Step 3: Examine the Remainder

Look at the expression inside the parentheses after factoring out the common factor: $$(y^2 + 3y + 2xy)$$ Now, let's identify common factors among the last two terms of this expression: We have a common factor of y in these terms.

Step 4: Factoring Out Common Factors in the Remainder

Now, let's factor out the common factor y from the last two terms: $$2(y^2 + y(3 + 2x))$$.

Step 5: Final Result

The final factored form of the polynomial is: $$2(y^2 + y(3 + 2x))$$

Key Concepts

Common Factors

In algebra, identifying common factors is an essential step in simplifying polynomials or any algebraic expressions. A common factor is simply a number or variable that can evenly divide each term in the expression.

Here's how to spot and deal with common factors:

• Check each term in the polynomial for any shared numbers or variables.
• The greatest number or highest power of a variable that all terms share is the common factor.
• Once identified, you can factor it out, simplifying the expression.

For the problem at hand, the polynomial is given as \( 2y^2 + 6y + 4xy \). We noted that '2' is a common factor across all terms. Factoring out this common factor gives us \( 2(y^2 + 3y + 2xy) \). Identifying and factoring common factors can greatly simplify the expression and make further factoring much more manageable.

Factoring Polynomials

Factoring polynomials is the process of breaking down a polynomial into products of simpler polynomials or factors. This process often starts with identifying common factors, but it can include other methods depending on the polynomial.

In this context:

• After factoring out the common factor "2" from the polynomial \(2y^2 + 6y + 4xy\), we are left with \(y^2 + 3y + 2xy\) inside the parentheses.
• The next step focuses on extracting factors within the remaining polynomial \(y^2 + 3y + 2xy\).
• The last two terms, \(3y + 2xy\), share a common factor "y".

Thus, we factor "y" out and reformat the expression, obtaining \(2(y^2 + y(3 + 2x))\). Through the meticulous process of identifying and factoring internal patterns, complex expressions are broken down into simpler segments.

Algebraic Expressions

Algebraic expressions are mathematical statements that can include numbers, variables, and operations. They can vary in complexity and size, but breaking them down is part of mastering algebra.

When factoring expressions like our given polynomial \(2y^2 + 6y + 4xy\):

• Separate the expression into terms - these are parts of algebraic expressions separated by addition or subtraction.
• Look for patterns or commonalities, such as shared numbers or variables that can be extracted.
• Remember that some expressions might need more than one round of factoring before they’re completely simplified.

In this problem, beginning with extracting numerical common factors and then delving deeper into internal terms illustrates the layered approach needed. Understanding algebraic expressions requires a step-by-step breakdown of terms to achieve fully factored forms.