Problem 61 - Beginning and Intermediate Algebra

Question

Factor completely. $$5 c^{2}+23 c d+12 d^{2}$$

Step-by-Step Solution

Answer

The completely factored form of the given expression is \((5c + 12d)(c + d)\).

Step 1: Rewrite the expression in group form

First, we identify two groups of terms sharing common factors. To do so, we will rearrange the terms to make the grouping more evident. In our case, the expression can be written as: \[(5c^2 + 12d^2) + (23cd)\]

Step 2: Factor out common factors from the two groups

Now we will factor out the common factors from the two groups of terms: \[c(5c + 12d) + d(23c)\]

Step 3: Identify the common binomial factor

Now we can see that there is a common binomial factor, which is \(5c + 12d\). So we can factor that common factor out of the expression: \[(5c + 12d)(c + d)\] The completely factored form of the given expression is \((5c + 12d)(c + d)\).

Key Concepts

Polynomial Expressions

Polynomial expressions are mathematical expressions that consist of variables, coefficients, and exponents combined using addition, subtraction, multiplication, and even division by a constant (but not division by a variable). These expressions can range from simple ones, like linear polynomials, to more complex ones, like quadratics and cubics. A polynomial is typically written in standard form, with terms arranged in descending order of their exponents.
Some key points to remember:

• A polynomial's degree is the highest power of the variable present in the expression.
• Constants can also be considered polynomials with a degree of zero.
• The coefficients, in mathematical terms, are the numerical factors that multiply each term's variable part.

Understanding polynomial expressions is crucial as they form the very basis for various algebraic operations, like factoring and simplifying.

Grouping Method

The grouping method is a technique used for factoring polynomials, especially when dealing with quadratic expressions that are not easily factorable through traditional means. Instead of handling the entire expression at once, the grouping method breaks the polynomial into smaller portions or groups, making it easier to identify common factors.
Here's how you can think about the process:

• First, divide the polynomial into groups that might share a common factor.
• Next, factor out the greatest common factor from each group.
• Then, look for a common factor across all the groups.

This method is effective because it simplifies complex expressions into more tractable components. Once groups are clearly defined, factoring becomes simpler, as you deal with smaller parts rather than the entire expression at once.

Common Factors

A common factor is a number or algebraic expression that divides two or more numbers or algebraic expressions evenly, leaving no remainder. In the context of polynomial expressions, identifying common factors is a critical step in the factoring process, as it simplifies the expression and makes further operations more manageable.
To find common factors, follow these tips:

• Look for variables and terms that appear in all parts of the polynomial.
• Consider the numerical coefficients and identify the highest factor common to those coefficients.
• Once found, write the common factor outside the parentheses, indicating it is factored out from each term.

Recognizing common factors not only aids in simplifying expressions but also prepares the polynomial for complete factorization.

Factoring Techniques

Factoring techniques are the different methods used to break down polynomials into simpler components, known as factors. This decomposition is essential in solving algebraic equations and simplifying expressions. Different techniques apply based on the structure and degree of the polynomial.
Some common factoring techniques include:

• **Factoring by grouping**: This is effective when there are four or more terms, and involves grouping terms to find common factors.
• **Greatest common factor (GCF)**: This technique involves identifying the largest factor shared by all terms in the polynomial.
• **Difference of squares**: Used when a polynomial can be expressed as the difference between two perfect squares.

Applying the appropriate factoring technique depends on the complexity and form of the polynomial, thus aiding in simplifying and solving polynomial equations efficiently.