Problem 18 - Beginning and Intermediate Algebra

Question

Complete the factorization. $$8 c^{2}+14 c d-9 d^{2}=(4 c+9 d)(\quad)$$

Step-by-Step Solution

Answer

The short answer to the question is: \(8c^2 + 14cd - 9d^2 = (4c + 9d)(2c - d)\).

Step 1: Set up the problem with the given factor

We know that one of the factors is (4c + 9d), so let's set up the problem as: \(8c^2 + 14cd - 9d^2 = (4c + 9d)(Af + Bd)\), where Af represents the coefficient of f (c or d) in the matching factor and Bd represents the coefficient of d in the matching factor.

Step 2: Multiply the given factor with the matching factor

Next, we will multiply the given factor (4c + 9d) with the matching factor (Af + Bd) using the distributive property (also known as FOIL method). So we have: \((4c + 9d)(Af + Bd) = 4c \cdot Af + 4c \cdot Bd+ 9d \cdot Af + 9d \cdot Bd\)

Step 3: Compare coefficients with the given trinomial

Now, let's compare the coefficients of the terms in the expression obtained in Step 2 with the coefficients in the given trinomial: 1. Coefficient of \(c^2\): \(4c \cdot Af = 8c^2\) => \(Af = 2\) 2. Coefficient of \(d^2\): \(9d \cdot Bd = -9d^2\) => \(Bd = -1\) 3. Coefficient of \(cd\): \(4c \cdot Bd + 9d \cdot Af = 14cd\) => \(4c \cdot (-1) + 9d \cdot 2 = 14cd\)

Step 4: Solve for the matching factor's coefficients

Using the relations found in Step 3, we can solve for the matching factor's coefficients: 1. Since Af = 2, our matching factor will be (2f + Bd) 2. Since Bd = -1, our matching factor will be (2f - d)

Step 5: Determine the variables f and d in the matching factor

Now, we need to determine what variables f and d should be in the matching factor (2f - d) such that it matches the term coefficients in the given trinomial. From the coefficients of cd term in step 3, we have: \(4c(-1) + 9d(2) = 14cd\) Simplifying it: \(-4c + 18d = 14cd\) Since this identity holds true, we can conclude that f = c and d = d in the matching factor, making it (2c - d).

Step 6: Write the final factorization

Now that we have the matching factor, we can write the final factorization: \(8c^2 + 14cd - 9d^2 = (4c + 9d)(2c - d)\)

Key Concepts

Distributive Property

The distributive property is a fundamental principle in algebra that allows us to multiply a sum by another number. To put it simply, when we have an expression like \(a(b + c)\), we can distribute the multiplication across the terms inside the parentheses. This means that \(a(b + c) = ab + ac\).
It's a useful tool, especially when dealing with polynomials or factorization problems like the one in the given exercise.
The distributive property ensures that each element inside the parentheses is multiplied separately by the term outside.

• In our context, we applied this property when multiplying \(4c + 9d\) by \(Af + Bd\).
• By distributing terms, we formed \(4c \cdot Af + 4c \cdot Bd + 9d \cdot Af + 9d \cdot Bd\).

This approach is crucial for breaking down complex algebraic expressions!

Trinomial

A trinomial is a type of polynomial that consists of exactly three terms. These terms are usually separated by a plus or minus sign, and each can be a combination of variables and constants.
In the original exercise, the expression \(8c^2 + 14cd - 9d^2\) is a trinomial. It has:

• The term \(8c^2\), involving the variable \(c\) squared.
• The term \(14cd\), involving both variables \(c\) and \(d\).
• The term \(-9d^2\), involving the variable \(d\) squared.

Understanding trinomials is vital for factorization as they are quite common in various algebraic problems.
Identifying each term's degree and coefficient helps in applying algebraic methods like the FOIL to factorize the expressions easily.

FOIL Method

The FOIL method is a specific technique used to expand binomials productively. FOIL stands for First, Outer, Inner, Last, which are the steps taken to multiply two binomials.
Our initial setup in the factorization exercise was to use the FOIL method on the binomials \(4c + 9d\) and \(Af + Bd\). Here's a breakdown:

• First: Multiply the first terms of each binomial, i.e., \(4c \cdot Af\).
• Outer: Multiply the outer terms, i.e., \(4c \cdot Bd\).
• Inner: Multiply the inner terms, i.e., \(9d \cdot Af\).
• Last: Multiply the last terms, i.e., \(9d \cdot Bd\).

Using FOIL, we construct an equivalent expanded polynomial. With careful application, it's straightforward to handle the coefficients and keep track of variable interactions.
It simplifies expanding binomials into a clear, step-by-step process.

Coefficient Comparison

Coefficient comparison is a valuable technique used to solve algebraic equations involving polynomials. In this method, we align terms with the same degree across both sides of an equation and compare their respective coefficients.
In our problem, we determined the coefficients of the polynomial \(8c^2 + 14cd - 9d^2\) by comparing it to its expanded form \(4c \cdot Af + 4c \cdot Bd + 9d \cdot Af + 9d \cdot Bd\).
By matching coefficients, we deduced:

• The coefficient for \(c^2\) led us to find \(Af = 2\).
• The coefficient for \(d^2\) led to \(Bd = -1\).
• For the mixed term \(cd\), the equation led to confirming the relation between \(Af\) and \(Bd\).

This helped us find the accurate factors needed, effectively simplifying the expression into \( (4c + 9d)(2c - d) \).
It's a logical approach to decipher complex polynomial relationships efficiently.