Problem 15 - Algebra for College Students

Question

Factor completely each of the polynomials and indicate any that are not factorable using integers. $$x^{2}+15 x y+36 y^{2}$$

Step-by-Step Solution

Answer

The polynomial factors to \((x+12y)(x+3y)\).

Step 1: Identify the Type of Polynomial

The given polynomial is a quadratic trinomial of the form \(ax^2 + bxy + cy^2\), where \(a=1\), \(b=15\), and \(c=36\). We will confirm if it can be factored into a pair of binomials.

Step 2: Apply the Factoring Method for Quadratic Trinomials

To factor \(x^2 + 15xy + 36y^2\), we need to find two numbers that multiply to \(ac=36\) (since \(a=1\) and \(c=36\)), and add up to \(b=15\). These numbers are 3 and 12.

Step 3: Rewrite Middle Term with Found Numbers

Rewrite the middle term \(+15xy\) using the numbers found: \(x^2 + 3xy + 12xy + 36y^2\). This breaks down the expression into four terms, making it easier to factor by grouping.

Step 4: Factor by Grouping

Group the terms: \((x^2 + 3xy) + (12xy + 36y^2)\). Factor out the greatest common factor from each group. From the first group: \(x(x + 3y)\). From the second group: \(12y(x + 3y)\). Now the expression is \(x(x + 3y) + 12y(x + 3y)\).

Step 5: Factor Out the Common Binomial Factor

Factor out the common binomial \((x + 3y)\) from the entire expression, resulting in \((x + 12y)(x + 3y)\). Therefore, this is the completely factored form of the polynomial.

Key Concepts

Quadratic Trinomials

Quadratic trinomials are polynomials with three terms, taking the form \(ax^2 + bxy + cy^2\). They are quadratic because the highest degree is two, generally represented as \(ax^2\) in more basic forms. Understanding how these trinomials work is key to mastering polynomial factorization.

Typically, you’ll identify a quadratic trinomial by observing:

• The first term is a squared term such as \(x^2\).
• The second term is a linear term featuring different variables, often written as \(bxy\).
• The third term is a squared term of another variable, like \(cy^2\).

Recognizing the pattern \(ax^2 + bxy + cy^2\) helps you apply appropriate factoring techniques to simplify or solve the equation. Key is focusing on the coefficients \(a\), \(b\), and \(c\), since they dictate the relationships within the polynomial.

Factoring by Grouping

Factoring by grouping is an effective method to factor polynomials that are not immediately obvious. It involves splitting the middle term into two terms to make four terms in total. This approach becomes very useful in polynomial expressions like quadratic trinomials.

To factor by grouping:

• First, look for two numbers whose product is equal to \(ac\) and whose sum is \(b\).
• Rewrite the middle term using these two numbers, effectively splitting it into two parts.
• Next, group terms in pairs, allowing you to factor out the greatest common factor (GCF) from each pair.
• Lastly, if both groups contain a similar binomial factor, factor out this binomial to achieve the completely factored expression.

Using this method on the given problem, we find that \(x^2 + 15xy + 36y^2\) is transformed to \((x^2 + 3xy) + (12xy + 36y^2)\), which reveals the common binomial when factored out.

Common Binomial Factor

The concept of a common binomial factor is integral to the rearrangement and simplification of polynomial expressions, especially after factoring by grouping. Identifying and extracting this factor simplifies the polynomial considerably.

A common binomial factor is a shared expression within each group that you've identified through the previous steps:

• After factoring by grouping, each group should ideally share a common binomial.
• This shared binomial is a factor, which when extracted, provides a streamlined expression.

In our exercise, after grouping, we derived \(x(x + 3y) + 12y(x + 3y)\). Here, the repeated binomial factor is \((x + 3y)\). Pulling it out gives a much simpler expression, resulting in \((x + 12y)(x + 3y)\). This step completes the factoring process and reveals the inherent structure within the polynomial.