Problem 5 - Elementary Algebra

Question

Use the grouping method to factor the following polynomials. $$ 40 a b x-24 a b x y-35 c^{2} x+21 c^{2} x y $$

Step-by-Step Solution

Answer

Question: Factor the given polynomial using the grouping method: \(40abx - 24abxy - 35c^2x + 21c^2xy\). Answer: \((5 - 3y)x(8ab - 7c^2)\).

Step 1: Arrange the terms in pairs.

We can start by arranging the given polynomial terms into pairs: $$ (40abx - 24abxy) + (-35c^2x + 21c^2xy) $$

Step 2: Factor out the greatest common factor in each pair.

Now we'll find the GCF (Greatest Common Factor) for each pair of terms and factor it out. For the first pair, the GCF is \(8abx\). So we have: $$ 8abx(5 - 3y) $$ For the second pair, the GCF is \(7c^2x\). So we have: $$ 7c^2x(-5 + 3y) $$ Now our polynomial looks like this: $$ 8abx(5 - 3y) + 7c^2x(-5 + 3y) $$

Step 3: Factor out the common binomial factor.

We can see that \((5 - 3y)\) is a common factor in both expressions. Now we'll factor it out: $$ (5 - 3y)(8abx - 7c^2x) $$

Step 4: Simplify the expression (Optional but recommended).

We can further factor out the common factor \(x\) from the second term in the parentheses: $$ (5 - 3y)x(8ab - 7c^2) $$ The polynomial is now factored using the grouping method, and the final factored form is: $$ (5 - 3y)x(8ab - 7c^2) $$

Key Concepts

Grouping Method

The grouping method is an effective technique used for factoring polynomials, especially when dealing with four terms. This method involves rearranging and factoring in smaller portions. First, you identify terms that can be grouped together. You arrange the polynomial into pairs or groups of terms. This allows you to factor each group separately.

• Carefully arrange the polynomial terms into pairs or groups.
• Find a common factor for each group and factor it out.
• Look for a common binomial factor in the resulting expression.

It serves as a great way to simplify algebraic expressions, making them easier to work with and solve. This method requires practice to recognize suitable groupings. However, once mastered, it becomes a powerful tool in polynomial factoring.

Greatest Common Factor

The greatest common factor (GCF) is an essential concept when it comes to factoring polynomials. It represents the largest factor shared by all terms within a group. To find the GCF, you identify the highest power of each variable that appears in every term, along with the largest numerical coefficient.

• Break down each term into its prime factors.
• Identify the smallest power of each factor common across all terms.
• Multiply these common factors together to find the GCF.

Using the GCF helps simplify expressions as it allows pulling out a common term from all components in the group. In the provided solution, the GCF is used in both pairs to simplify them further before identifying a common binomial factor.

Binomials

Binomials play a crucial role in algebra and polynomial expressions. They consist of two terms separated by a plus or minus sign. In factoring, especially using the grouping method, recognizing a common binomial factor can allow further simplification of the expression.
For example, given polynomials might appear with common binomial factors like \((a - b)\) or \((x + y)\).

• A binomial is in the form \(ax + by\).
• Look for common binomial factors during factoring.
• Factor them out to further reduce the polynomial expression.

Identifying these repeating patterns helps simplify complex polynomial expressions. They allow for a reduction that might not be immediately apparent, aiding in streamlining expressions for easier manipulation or solving.

Polynomial Expressions

Polynomials are expressions made up of variables and coefficients, involving operations like addition, subtraction, and multiplication. They form the foundation of algebra and come in various forms, such as monomials, binomials, and trinomials.
Key Characteristics:

• Variable terms are raised to non-negative integer powers.
• The degree of a polynomial is determined by the highest power of its variables.
• Simplifying involves factoring out common terms and applying methods like the grouping method.

Understanding polynomial expressions is crucial for manipulating and solving algebraic problems. Recognizing the structure and form of polynomials helps you decide on the best approach for factoring them, such as using the GCF or other factoring techniques.