Problem 57 - Algebra for College Students

Question

Factor by grouping. $$2 a c+3 b d+2 b c+3 a d$$

Step-by-Step Solution

Answer

The expression factors to \((a + b)(2c + 3d)\).

Step 1: Group Terms

The expression to be factored is \(2ac + 3bd + 2bc + 3ad\). Start by grouping the terms in such a way that there is a common factor in each group. We can group the terms as follows: \((2ac + 3ad) + (2bc + 3bd)\).

Step 2: Factor Out Common Factors

Look at each grouped pair of terms and factor out the greatest common factor (GCF). - For the first group \((2ac + 3ad)\), the GCF is \(a\), so factor out \(a\) to get \(a(2c + 3d)\).- For the second group \((2bc + 3bd)\), the GCF is \(b\), so factor out \(b\) to get \(b(2c + 3d)\).

Step 3: Recognize a Common Binomial Factor

Notice that both terms \(a(2c + 3d)\) and \(b(2c + 3d)\) now contain the common binomial factor \((2c + 3d)\). Factor this common binomial out of the expression.

Step 4: Factor Out the Common Binomial

Take the common factor \((2c + 3d)\) out of the entire expression: \((a + b)(2c + 3d)\). This is the fully factored form of the expression.

Key Concepts

Greatest Common Factor

The greatest common factor (GCF) is the largest factor shared by all terms in a group. It plays a crucial role when factoring expressions, as it allows us to simplify the expression by dividing out the GCF from each term.
To find the GCF:

• List the factors of each term in the group.
• Identify the highest common factor shared by all.

For example, in the term grouping \(2ac + 3ad\), each term \(2ac\) and \(3ad\) contains an \(a\), which is the GCF. By factoring out \(a\), we simplify the expression to \(a(2c + 3d)\).
This step is repeated for other groupings as well, helping simplify larger algebraic expressions and setting the stage for further factoring.

Algebraic Expressions

Algebraic expressions consist of variables, numbers, and operators. These expressions can often appear complex, but breaking them down into smaller parts or terms can make them easier to handle.
When focusing on algebraic expressions:

• Identify individual terms, which are linked by addition or subtraction.
• Look for shared factors or structures among terms to simplify them.

In the expression \(2ac + 3bd + 2bc + 3ad\), the terms are combined with addition, and we can look for patterns or common factors. By grouping terms such as \(2ac\) and \(3ad\) separately from \(2bc\) and \(3bd\), we can identify and factor out these patterns efficiently.
This breakdown process transforms complex polynomial expressions into simpler components, aiding in understanding and manipulation.

Polynomial Factoring

Polynomial factoring involves breaking down a polynomial into simpler components, which, when multiplied together, give back the original expression. It is a fundamental aspect of simplifying and solving algebraic equations.
Steps for effective polynomial factoring:

• Group terms in the polynomial that might share a common factor.
• Factor out common elements from these groups, such as shared variables or coefficients.
• Recognize common factors across the entire expression or identify common binomial factors.

In our example, after grouping and factoring, we find the expression \(a(2c + 3d) + b(2c + 3d)\). This reveals a common binomial \(2c + 3d\), which can be factored out to yield the final factorized form: \((a + b)(2c + 3d)\).
Understanding this method helps solve and simplify polynomial expressions, crucial in both basic algebra and more advanced mathematics.