Problem 62 - Elementary and Intermediate Algebra

Question

Factor. If an expression is prime, so indicate. $$ 8 c^{2}-10 c d+3 d^{2} $$

Step-by-Step Solution

Answer

The factored form is \((4c - 3d)(2c - d)\).

Step 1: Identify the expression structure

The given expression is a quadratic trinomial: \(8c^2 - 10cd + 3d^2\). Recognize that it is of the form \(ax^2 + bxy + cy^2\) with \(a = 8\), \(b = -10\), and \(c = 3\).

Step 2: Find two numbers whose product is \(a \, c\) and sum is \(b\)

Calculate the product \(ac = 8 \times 3 = 24\). We need to find two numbers that multiply to 24 and add to \(-10\). The numbers are \(-4\) and \(-6\).

Step 3: Rewrite middle term using found numbers

Rewrite the middle term \(-10cd\) using \(-4cd\) and \(-6cd\): \[ 8c^2 - 4cd - 6cd + 3d^2 \]

Step 4: Factor by grouping

Group the terms: \( (8c^2 - 4cd) + (-6cd + 3d^2) \). Factor the first group: \(4c(2c - d)\). Factor the second group: \(-3d(2c - d)\).

Step 5: Factor out the common binomial

Notice that \((2c - d)\) is common in both groups. Factor out \(2c - d\):\[ (4c - 3d)(2c - d) \]

Step 6: Verification

Multiply \((4c - 3d)(2c - d)\) to verify the factorization: Begin with distributing:\[= 4c(2c - d) - 3d(2c - d) \]Expanding gives \(8c^2 - 4cd - 6cd + 3d^2\), which matches the original expression.

Key Concepts

Quadratic Trinomials

Quadratic trinomials are an essential concept in algebra, often encountered by students. A trinomial is a polynomial consisting of three terms. When you have a quadratic trinomial, these three terms are usually structured as \(ax^2 + bx + c\). Here, \(a\), \(b\), and \(c\) are numbers or coefficients, and \(x\) is the variable raised to different powers.

Understanding the structure of quadratic trinomials is crucial because it lays the groundwork for factoring these expressions. Recognizing the standard form helps in identifying suitable methods for factorization. For example, in the expression \(8c^2 - 10cd + 3d^2\), it follows the form \(ax^2 + bxy + cy^2\), where each term consists of products of coefficients and variables.

To factor a quadratic trinomial effectively, you first need to grasp the idea that it represents a parabola if graphed. Many quadratic trinomials can be factored into the product of two binomials, which is often helpful in solving equations or simplifying expressions. By practicing to identify and rearrange terms, factoring becomes more intuitive and manageable.

Polynomial Expressions

Polynomial expressions are mathematical statements comprising variables and coefficients. These expressions can include addition, subtraction, and multiplication, but only involve non-negative integer exponents of variables.

Polynomials are categorized based on the number of terms they contain:

• Monomials have one term, such as \(3x\).
• Binomials contain two terms, like \(4x + 5\).
• Trinomials, like the quadratic trinomial \(8c^2 - 10cd + 3d^2\), include three terms.

The degree of a polynomial is determined by the highest power of the variable present. For instance, a quadratic trinomial has a degree of 2 because the highest exponent of the variable is 2. Understanding this allows us to anticipate the structure and behavior of polynomials during operations like addition and multiplication.

Factoring polynomial expressions involves expressing them as a product of simpler polynomials. This process plays a vital role in algebraic manipulation, allowing us to simplify expressions, solve equations, and explore the properties of algebraic functions.

Factoring by Grouping

Factoring by grouping is a technique used to factor polynomials, especially when dealing with trinomials or polynomials with four terms. This method is particularly handy when traditional factoring techniques, like extracting a greatest common divisor (GCD), are not directly applicable.

In the given exercise, the expression \(8c^2 - 10cd + 3d^2\) is rearranged to allow for grouping. The idea is to strategically rearrange and group terms so that each group has a common factor. Here’s a simplified breakdown of the process:

• First, arrange the expression to create two groups, like \( (8c^2 - 4cd) + (-6cd + 3d^2) \).
• Next, factor out the common factor from each pair of terms: \(4c(2c - d)\) and \(-3d(2c - d)\).
• Observe the common binomial, \(2c - d\), in each group.
• Finally, factor the common binomial out, resulting in \((4c - 3d)(2c - d)\).

This technique turns a seemingly complicated expression into a product of simpler factors. Factoring by grouping reinforces recognition of structure within polynomials, providing a powerful tool for breaking down complex expressions. Practicing this method enhances problem-solving skills and aids in understanding more advanced algebraic concepts.