Problem 56 - Beginning and Intermediate Algebra

Question

Factor by grouping. $$c d-5 d+8 c-40$$

Step-by-Step Solution

Answer

The expression \(cd-5d+8c-40\) can be factored by grouping into \((c-5)(d+8)\).

Step 1: Identify the terms to group

In the given expression, we can see the terms can be grouped into two pairs: $$cd-5d$$ and $$8c-40$$

Step 2: Factor out the greatest common factors from each group

Now, we find the greatest common factors for each pair and factor them out: For the pair \(cd-5d\), the greatest common factor is \(d\). Factoring out \(d\), we get: $$d(c-5)$$ For the pair \(8c-40\), the greatest common factor is \(8\). Factoring out \(8\), we get: $$8(c-5)$$

Step 3: Combine the factored pairs

Now that we have factored out the greatest common factors from each group, we can combine the factored pairs: $$d(c-5) + 8(c-5)$$

Step 4: Factor by grouping

Notice that there is a common factor of \((c-5)\) in both terms. Factoring this out, we get: $$(c-5)(d+8)$$ Therefore, the expression \(cd-5d+8c-40\) can be factored by grouping as \((c-5)(d+8)\).

Key Concepts

Greatest Common Factor

The greatest common factor (GCF) is the largest factor that divides two or more numbers or terms. It's a crucial step in simplifying expressions or solving equations.
When factoring expressions, the GCF helps to break up the expression into simpler terms. To find the GCF, list all the factors of each term and identify the largest factor they share.

• For example, in the terms \(cd\) and \(-5d \), \(d\) is the GCF because it's common in both terms.
• In \(8c \) and \(-40\), the largest common factor is \(8\).

By factoring out the GCF, you simplify the terms and prepare them for easier manipulation in algebraic expressions.

Algebraic Expressions

An algebraic expression is a mathematical phrase that includes numbers, variables, and operations. It's like a sentence that conveys a mathematical idea.
Expressions can include:

• Variables: letters that represent unknown values, like \(c\) and \(d\).
• Constants: fixed values such as \(-5\) or \(8\).
• Operators: symbols like addition or subtraction that indicate operations between terms.

In the expression \(cd - 5d + 8c - 40\), each group of numbers and variables forms a part of the whole puzzle you need to solve by factoring or simplifying.

Polynomials

Polynomials are a type of algebraic expression that consists of variables raised to whole number powers and their coefficients.
The expression \(cd - 5d + 8c - 40\) is considered a polynomial because it includes multiple terms added or subtracted together.

• Each part of the polynomial, like \(cd\) or \(-5d\), is called a "term."
• The degree of a polynomial term is determined by the sum of the powers of the variables in it. This particular expression doesn't have explicit powers visible but is implied to be first-degree due to its simple variable format.

Polynomials can be manipulated using different techniques like factoring, which helps to solve equations or simplify expressions.

Factoring Techniques

Factoring is a powerful technique in algebra used to express an expression as a product of its factors. This process involves breaking down complex expressions into simpler, more manageable parts.
There are several factoring techniques:

• Factoring by Grouping: This strategy is useful when you have four terms or more. You group terms to find a common factor in each pair, as seen in the expression \(cd - 5d + 8c - 40\).
• Finding the Greatest Common Factor: As already explained, this is finding the largest factor shared by all terms.
• Special Factorizations: Such as factoring quadratics, perfect squares, and the differences of squares.

In the example, once the expression is grouped and the common factors are extracted, you can re-combine to simplify or solve the expression. This reinforces the importance of identifying patterns and factors within algebraic expressions to effectively simplify or solve them.