Problem 112 - Elementary Algebra

Question

Factor each trinomial of the form \(x^{2}+b x y+c y^{2}\). \(c^{2}-7 c d+18 d^{2}\)

Step-by-Step Solution

Answer

The factored form is \((c + 2d)(c - 9d)\).

Step 1: Identify coefficients

Identify the coefficients in the given trinomial. For the trinomial \(c^{2}-7cd+18d^{2}\), we have: \(a = 1\) (coefficient of \(c^{2}\)), \(b = -7\) (coefficient of \(cd\)), and \(c = 18\) (coefficient of \(d^{2}\)).

Step 2: Find two numbers that multiply to ac and add to b

We need to find two numbers that multiply to \(a \times c = 1 \times 18 = 18\) and add to \(b = -7\). These numbers are \(-9\) and \(2\) because \(-9 \times 2 = 18\) and \(-9 + 2 = -7\).

Step 3: Split the middle term

Rewrite the middle term \(-7cd\) using \(-9cd\) and \(+2cd\): \[c^{2} - 9cd + 2cd + 18d^{2}\].

Step 4: Factor by grouping

Group the terms in pairs and factor out the common factors: \[c(c - 9d) + 2d(c - 9d)\].

Step 5: Factor out the common binomial

Factor out the common binomial factor \(c - 9d\): \[(c + 2d)(c - 9d)\].

Key Concepts

Understanding Trinomial Factoring

Trinomial factoring is a technique used to break down a polynomial with three terms into simpler factors. This helps in solving equations and simplifying algebraic expressions. Let’s look into a specific example to understand the concept.

Consider the trinomial: \(c^{2}-7cd+18d^{2}\).
We want to express this as the product of two binomials. The steps for factoring trinomials are:

• Identify the coefficients: Here, the coefficients are \(a = 1\) for \(c^{2}\), \(b = -7\) for \(cd\), and \(c = 18\) for \(d^{2}\).
• Find two numbers that multiply to the product of \(a \times c = 1 \times 18 = 18\) and add to \(b = -7\).
• Rewrite the middle term using these two numbers.
• Group terms and factor out common factors.
• Extract the common binomial factor.

Following these steps effectively simplifies complex expressions.

Understanding Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations. They form the backbone of algebra problems. When dealing with expressions like \(c^{2}-7cd+18d^{2}\), understanding the structure is crucial.

In our example, we combine terms involving the same variables, like \(c^{2}\) and \(d^{2}\). Here’s a breakdown:

• {{}}\(c^{2}\): This is the quadratic term with variable \(c\).
• {{}}\(-7cd\): The linear term involving both \(c\) and \(d\).
• {{}}\(18d^{2}\): The quadratic term with variable \(d\).

Recognizing and categorizing these components helps us in rearranging and factoring the expressions efficiently. Clear understanding makes problem-solving easier.

Polynomial Factorization Techniques

Polynomial factorization is the process of breaking down a polynomial into a product of simpler polynomials. This simplification can make solving equations, integrating, or deriving much easier. Let’s look at a common technique used in our example.

Factoring by grouping involves:

• Splitting the middle term: Rewriting the trinomial \(c^{2}-7cd+18d^{2}\) by finding numbers that multiply to \(18\) and add to \(-7\).
• Grouping terms: The expression \(c^{2} - 9cd + 2cd + 18d^{2}\) is grouped into pairs: \(c(c - 9d) + 2d(c - 9d)\).
• Factoring out common binomials: Extracting the binomial \(c - 9d\) results in \((c + 2d)(c - 9d)\).

Mastering this method is key to tackling complex polynomials. Practice and familiarity with these techniques greatly enhance algebraic problem-solving skills.