Problem 13 - Elementary Algebra

Question

For the following problems, factor the polynomials. $$ 21 y-28 $$

Step-by-Step Solution

Answer

Answer: The factored form of the polynomial is $$7(3y - 4)$$.

Step 1: Identify the GCD of the polynomial terms.

For the given polynomial, $$21y - 28$$, we need to find the greatest common divisor of both the terms. We can list down the factors of each term: - Factors of $$21y$$ are 1, 3, 7, $$y$$, $$3y$$, $$7y$$, and $$21y$$. - Factors of $$28$$ are 1, 2, 4, 7, 14, and 28. Finding the greatest common divisor between them, we find it to be 7.

Step 2: Factor out the GCD from the polynomial.

Now that we have identified the GCD, we can factor it out from the polynomial. Factoring out $$7$$ from each term, we get: $$ 7(3y) - 7(4) $$

Step 3: Rewrite the polynomial with the factored GCD.

Now that we have factored out the GCD, $$7$$, we can rewrite the polynomial as: $$ 7(3y - 4) $$ So, the factored polynomial is $$7(3y - 4)$$.

Key Concepts

Greatest Common Divisor

The greatest common divisor (GCD) is a crucial concept in simplifying polynomials. In simple terms, the GCD of two numbers or terms is the largest number that divides both of them without leaving a remainder.
When working with algebraic expressions, the GCD helps us in identifying the common factor that can be factored out from the terms of the polynomial. To find the GCD for a polynomial, follow these steps:

• List the factors of each polynomial term separately.
• Identify the largest factor that appears in all lists as the common divisor.

Finding the GCD allows us to simplify polynomials effectively, making further algebraic operations much easier. Remember, it’s about finding that one number which every term in the expression can share.

Algebraic Expressions

Algebraic expressions are the building blocks of algebra. They consist of numbers, variables, and arithmetic operations. In the exercise given, the expression was a simple term combination: \(21y - 28\).
Each expression can include:

• Coefficients: Numbers multiplying the variables (like 21 in \(21y\)).
• Variables: Symbols that represent unknown values (such as \(y\)).
• Constants: Standalone numbers not linked with variables (such as 28).

Understanding algebraic expressions entails recognizing these parts and how they interact. This foundational knowledge assists in manipulating and simplifying expressions when solving equations, such as through factoring.

Polynomial Factoring

Polynomial factoring involves breaking down a polynomial into a product of simpler terms or factors. It's a core skill in algebra that is used for simplifying expressions and solving equations.
The process of polynomial factoring generally involves:

• Identifying the greatest common divisor (GCD) of the polynomial's terms.
• Dividing each term by the GCD to simplify the expression.
• Rewriting the expression as a product of its factors.

In the given example, after identifying the GCD of 7 from \(21y - 28\), we factor out 7 to rewrite it as \(7(3y - 4)\).
This transformation makes the polynomial easier to work with for further algebraic operations and solutions.