Problem 85 - Algebra and Trigonometry with Analytic Geometry

Question

Factor the polynomial. $$ 2 a x-6 b x+a y-3 b y $$

Step-by-Step Solution

Answer

The factored polynomial is \((a - 3b)(2x + y)\).

Step 1: Identify Common Factors

First, let's look at the polynomial: \[ 2ax - 6bx + ay - 3by \]. Observe the terms to find the common factors in the groups: \[ 2ax - 6bx \] and \[ ay - 3by \].Clearly, 'x' is a common factor in the first group, and 'y' is a common factor in the second group. The coefficients each share a divisor with other terms:- From \(2ax - 6bx\), the factor is \(2x\).- From \(ay - 3by\), the factor is \(y\).

Step 2: Factor by Grouping

Now, apply factoring by grouping. Start with the first group:\[ 2ax - 6bx = x(2a - 6b) \] Then, factor the second group:\[ ay - 3by = y(a - 3b) \]. Now our expression becomes:\[ x(2a - 6b) + y(a - 3b) \].

Step 3: Simplify the Expression

Notice that each grouped expression has a similar internal factor. The expression inside the parentheses for the second term partly simplifies:\[ 2a - 6b \] simplifies to \(2(a - 3b)\).Therefore, rewrite our original expression:\[ x(2(a - 3b)) + y(a - 3b) \].

Step 4: Extract the Common Factor

Having rewritten the expression, observe that both grouped terms include \((a - 3b)\) as a factor:Now factor out \((a - 3b)\):\[ (a - 3b)(2x + y) \]. This is now the factored form of the original polynomial.

Key Concepts

Factor by Grouping

Factor by grouping is a fantastic method to simplify polynomials by rearranging and grouping terms that share common factors. In this method, you first look for terms that can be conveniently grouped into smaller expressions.
After identifying these groups, factor out the common terms within each group.

• For instance, observe the polynomial: \(2ax - 6bx + ay - 3by\).
• The idea is to group terms as \((2ax - 6bx)\) and \((ay - 3by)\).
• In the first group, \(x\) is a common factor, and in the second group, \(y\) is common.

By rearranging in this way and factoring from each group, you carve a path to simplify the polynomial effectively.
It's like organizing your bookshelf by genres, making each section neat and coherent for easy access.
Once you master this method, complex polynomials become more approachable, and solving them gets easier.

Common Factors

Common factors are key to simplifying expressions and play a crucial role in factoring polynomials.

• A common factor is simply a term or number that divides two or more numbers or terms without leaving a remainder.
• To identify them, take a close look at the coefficients and variables in each group.
• For \(2ax - 6bx\), the coefficient 2 in \(2ax\) and 6 in \(6bx\) share 2 as a divisor. Similarly, both terms contain \(x\).
• In \(ay - 3by\), only the \(y\) variable is common.

Recognizing these commonalities enables you to simplify expressions accurately.
They act as the thread that lets you unravel a complex polynomial into a simpler, more understandable form.
Factor out these common elements to build an intermediary step towards further simplification, greatly aiding in finding the most simplified polynomial expression.

Polynomial Simplification

Simplifying a polynomial is about reducing it to its simplest, most compact form.
This involves several steps, like factoring out common elements and using factoring techniques like grouping. First, identify any internal simplifications:

• In our case, \(2a - 6b\) can be simplified to \(2(a - 3b)\).
• This simplification is pivotal because it harmonizes with the second grouped term \((a - 3b)\).
• By acknowledging this similarity between the terms, you can further factor it down to \((a - 3b)(2x + y)\).

This shows how the expression can be reduced to fewer terms, effectively streamlining its appearance and making it easier to work with. Breaking down a polynomial into its simplest form is like chiseling a rough stone into an elegant sculpture.
It's a systematic approach involving finding hidden patterns and performing basic arithmetic and algebraic actions.