Problem 25 - College Algebra with Modeling and Visualization

Question

Use grouping to factor the polynomial. \(y^{4}+2 y^{3}-5 y^{2}-10 y\)

Step-by-Step Solution

Answer

The polynomial is factored as \((y^3 - 5y)(y + 2)\).

Step 1: Identify Grouping Pairs

Start by dividing the polynomial into two pairs that we'll work with separately. The given polynomial is \(y^{4} + 2y^{3} - 5y^{2} - 10y\). Group the terms as follows: \((y^{4} + 2y^{3}) + (-5y^{2} - 10y)\).

Step 2: Factor Out Common Factor from Each Group

Look inside each group and factor out the greatest common factor (GCF). In the first group, \(y^{4} + 2y^{3}\), the GCF is \(y^3\), so we get \(y^3(y + 2)\). In the second group, \(-5y^{2} - 10y\), the GCF is \(-5y\), giving us \(-5y(y + 2)\).

Step 3: Factor by Grouping

Notice that both groups now contain the common polynomial factor \((y + 2)\). Factor \((y + 2)\) out of the entire expression, resulting in \((y^3 - 5y)(y + 2)\).

Step 4: Check Your Work

Expand the factored form \((y^3 - 5y)(y + 2)\) to verify it matches the original polynomial. \(y^3(y + 2) - 5y(y + 2)\) simplifies back to \(y^4 + 2y^3 - 5y^2 - 10y\). This confirms that the factorization is correct.

Key Concepts

Grouping Method

The grouping method is a useful technique for factoring polynomials, especially when dealing with four-term polynomials like the one in our exercise. When employing the grouping method:

• We divide the polynomial into two segments or groupings of terms.
• Next, we focus on each segment separately, finding a way to factor each one.
• The goal is to find a common factor between these two groups, which can then be factored out.

This method is particularly effective when direct factoring isn't immediately apparent. Take our polynomial, for example, the expression is split into two groups: y^{4} + 2y^{3} and -5y^{2} - 10y. A keen observation of these groupings reveals potential commonalities that can make factorization more manageable.

Greatest Common Factor

The Greatest Common Factor (GCF) is a key concept in algebra and is essential when factoring polynomials. It involves identifying the largest expression that divides all terms in a particular group without leaving a remainder. Here’s how it works:

• Examine each group separately and determine their respective GCF.
• In our example, the first group, consisting of y^{4} + 2y^{3}, has a GCF of y^3.
• The second group, -5y^{2} - 10y, features a GCF of -5y.

Once the GCFs are factored out, we are left with two expressions that have a common polynomial factor, a crucial step for successful factorization via grouping. Recognizing and extracting the GCF simplifies the polynomial, paving the way for efficient factorization.

Algebra

Algebra is a branch of mathematics where symbols, often letters, represent numbers or quantities in equations and formulas. Understanding algebra is essential for mastering techniques such as polynomial factorization. When dealing with expressions like those in our example:

• It is important to understand the properties of exponents and coefficients.
• Algebraic manipulation is required to regroup terms and simplify expressions.
• Being familiar with terms allows us to easily spot potential factors and common elements.

Through algebra, we can identify patterns and develop strategies for simplifying complex expressions into more accessible factored forms. It’s a fundamental skill that supports various mathematical procedures, making it a vital component of mathematical education.

Factored Form

Factored form in algebra refers to the expression of a polynomial as a product of its factors. This form is simpler and provides insight into the roots or solutions of equations derived from these polynomials. Achieving factored form involves:

• Identifying and explaining patterns or common factors in the polynomial.
• Manipulating expressions to extract these factors clearly.
• The factored form derived from our exercise is (y^3 - 5y)(y + 2). This shows how the original expression can be expressed in a product of simpler binomial or monomial factors.

Factored form is not only a practical representation but also a powerful analysis tool. It reveals the underlying structure of polynomials, aids in simplifying mathematical problem-solving, and is a stepping stone to finding polynomial roots.