Problem 8 - Beginning and Intermediate Algebra

Question

Factor by grouping. $$ 10 y^{2}-8 y z-15 y z+12 z^{2} $$

Step-by-Step Solution

Answer

The factored form of the given expression, \(10y^2 - 8yz - 15yz + 12z^2\), by grouping is \((5y - 4z)(2y - 3z)\).

Step 1: Group the terms

First, let's group the terms in pairs: $$ (10y^2 - 8yz) + (-15yz + 12z^2) $$

Step 2: Factor out the common factors from each group

Now, let's factor out the common factor from each group: For the first group, both terms have a common factor of \(2y\): $$ 2y(5y - 4z) $$ For the second group, both terms have a common factor of \(-3z\): $$ -3z(5y - 4z) $$ Now our expression looks like this: $$ 2y(5y - 4z) - 3z(5y - 4z) $$

Step 3: Factor out the common binomial

We can now see that both terms have a common binomial factor, \((5y - 4z)\). We can factor it out: $$ (5y - 4z)(2y - 3z) $$ So, the factored form of the given expression is: $$ (5y - 4z)(2y - 3z) $$

Key Concepts

Polynomial Factoring

In algebra, polynomial factoring is an essential skill that simplifies complex expressions into a product of simpler factors. When you factor a polynomial, you're essentially finding two or more expressions that multiply together to recreate the original polynomial. This is a common technique used to solve equations, making them more manageable.

For example, consider the polynomial: \(10y^2 - 8yz - 15yz + 12z^2\). Through polynomial factoring, we transformed it into \((5y - 4z)(2y - 3z)\).

• **Step 1**: Look for ways to group terms that could have common factors.
• **Step 2**: Identify and extract common factors from each group.

Factoring polynomials by grouping is a specific technique where we group terms in pairs and factor out the greatest common factor from each. This reveals a common binomial factor that, when factored out, perfectly decomposes the polynomial into simpler expressions.
The goal of polynomial factoring is not only simplification but also to identify roots of the polynomial through setting the factors equal to zero.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and operation symbols. They can represent real-world scenarios in a simplified manner. An expression is built from a combination of terms, which are separated by plus or minus signs.

In the exercise above, the algebraic expression \(10y^2 - 8yz - 15yz + 12z^2\) consists of four distinct terms. Each term is composed of constants (numbers) and variables (letters).

Let's break down the expression:

• **10y²**: This term has a coefficient of 10 and involves the square of variable \(y\).
• **-8yz and -15yz**: These terms involve the product of variables \(y\) and \(z\), with coefficients of -8 and -15 respectively.
• **12z²**: This term includes the square of variable \(z\) with a coefficient of 12.

Grouping terms in expressions allows us to strategically prepare for factoring by enabling us to identify potential common factors. This separation simplifies complex algebraic expressions and identifies binomial terms for factoring out.

Binomial Factors

Finding binomial factors involves identifying two-term expressions that serve as a common element in each group of the original polynomial. A binomial factor is an expression made up of two terms, linked by a plus or minus sign, which can be pulled out from the polynomial.

In our exercise, after grouping and factoring out common factors from each grouped term, we identified the binomial \((5y - 4z)\) as a common factor. Notice how this binomial factor repeats in the expression after factoring the grouped terms:

\(2y(5y - 4z) - 3z(5y - 4z)\).

• **Common Binomial Factor**: The repeating binomial \((5y - 4z)\) acts as the linking factor, allowing us to factor it out completely.
• **Final Factored Form**: It results in the product \((5y - 4z)(2y - 3z)\), representing a fully factored expression of the original polynomial.

Recognizing and extracting binomial factors involves understanding the structure of algebraic expressions and the relationships between terms to efficiently simplify them. This process not only makes solving equations easier but also highlights potential solutions based on the properties of binomials.