Problem 80 - Algebra A Combined Function

Question

Factor out the GCF from each polynomial. Then factor by grouping. $$ 16 x^{2}+4 x y^{2}+8 x y+2 y^{3} $$

Step-by-Step Solution

Answer

The polynomial factors to \[2(2x + y)(4x + y^2)\].

Step 1: Identify the GCF

To factor out the Greatest Common Factor (GCF), identify the common factor in each term of the polynomial. The terms are \(16x^2\), \(4xy^2\), \(8xy\), and \(2y^3\). The GCF here is 2.

Step 2: Factor Out the GCF

Divide each term by the GCF, 2: \(16x^2 \div 2 = 8x^2\), \(4xy^2 \div 2 = 2xy^2\), \(8xy \div 2 = 4xy\), \(2y^3 \div 2 = y^3\). The polynomial becomes: \[ 2(8x^2 + 2xy^2 + 4xy + y^3) \].

Step 3: Regroup the Polynomial

Group the terms into pairs for easier factorization. First pair: \(8x^2 + 2xy^2\).Second pair: \(4xy + y^3\).

Step 4: Factor Each Group

For the first pair \(8x^2 + 2xy^2\), factor out their GCF, which is \(2x\). This gives \(2x(4x + y^2)\).For the second pair \(4xy + y^3\), factor out their GCF, \(y\), resulting in \(y(4x + y^2)\).

Step 5: Combine the Factored Terms

Notice that both groups contain \(4x + y^2\). Factor out \(4x + y^2\) from the expression: \[ 2x(4x + y^2) + y(4x + y^2) = (2x + y)(4x + y^2) \].

Step 6: Final Expression with the GCF

Remember to include the previously factored GCF, 2, into the expression and the final form of the polynomial is: \[ 2(2x + y)(4x + y^2) \].

Key Concepts

Greatest Common Factor

The greatest common factor (GCF) is an essential concept when dealing with polynomials. It helps simplify expressions by identifying the largest factor that is present within all terms of the polynomial.
The process begins by examining all the coefficients and variables in each term. You want to find the highest number that divides all coefficients, as well as the highest power of any shared variable.
For instance, in the expression \(16x^2 + 4xy^2 + 8xy + 2y^3\), you identify the coefficients \(16, 4, 8, \) and \(2\). The GCF here is 2.
The next step involves factoring out this GCF, resulting in each term being divided by the identified factor. The polynomial then becomes \(2(8x^2 + 2xy^2 + 4xy + y^3)\). Identifying and factoring out the GCF simplifies later processes, making polynomial manipulation much easier.

Grouping Method

The grouping method is a technique used to factor polynomials that have more than three terms. It effectively "groups" terms into smaller, more manageable pairs.
In our problem, after factoring out the GCF, the polynomial becomes \(8x^2 + 2xy^2 + 4xy + y^3\).
This step involves arranging the terms into pairs or groups that can each be simplified through their own common factors.

• The first group might be \(8x^2 + 2xy^2\).
• The second group could be \(4xy + y^3\).

Each group is then independently factored. This ensures none of the potential common factors between groups is overlooked. Grouping is especially handy when direct factoring isn't obvious, as it subdivides the expression into smaller, easily factorable sections.

Factorization Techniques

Factorization techniques include a variety of methods to simplify polynomials. They break down complex expressions into their basic building blocks.
Some common techniques include:

• Finding the GCF: Begin by identifying and dividing all terms by the largest common factor, as seen in the first step of this process.
• Factoring by grouping: After the GCF is factored out, separately group and then factor each subset of the expression.
• Recognizing patterns: Patterns such as difference of squares or perfect square trinomials can simplify factorization.

For the given problem, the focus is on finding the GCF and then using grouping. This breakdown builds a foundation for handling more intricate polynomial equations, where multiple methods may be required to see the full solution.

Algebra

Algebra is the field of mathematics that combines numbers and symbols to represent expressions and solve equations. It provides the foundational skills required for working with polynomials.
In algebra, factorization transforms expressions to reveal their simplest forms or solve equations. It is a critical skill for algebraic manipulation.
Working with polynomials, particularly through factoring, involves:

• Simplifying expressions: Breaking polynomials into simpler components can make solving equations more straightforward.
• Solving equations: By factoring, one can set each factor equal to zero to find solutions to polynomial equations.
• Understanding structure: Recognizing how components of an expression interact can deepen understanding of mathematical relationships.

Algebra goes beyond basic operations, permitting a more nuanced approach to problem-solving across various topics. By mastering these topics, students not only simplify and solve polynomial expressions but also gain confidence in handling broader mathematical challenges.