Problem 4 - Algebra 2

Question

Find the GCF of each expression. Then factor the expression. $$ 5 t^{2}+7 t $$

Step-by-Step Solution

Answer

The GCF of the expression is \(t\), and the factored expression is \(t(5t + 7)\).

Step 1: Determine the GCF

Identify the Greatest Common Factor (GCF) of the terms in the expression. The terms \(5t^2\) and \(7t\) both have \(t\) as a common factor. They do not have a numerical factor in common, so the GCF is \(t\).

Step 2: Factor out the GCF

Rewrite the expression by factoring out the GCF. The original expression is \(5t^2 + 7t\). After factoring out the GCF \(t\), the expression becomes \(t(5t + 7)\).

Key Concepts

Greatest Common Factor

When we talk about the Greatest Common Factor (GCF), we refer to the largest factor that divides two or more numbers or terms. In algebra, identifying the GCF is crucial for simplifying expressions and solving equations. To find the GCF of an algebraic expression, look at both the numerical coefficients and any common variables across the terms.

For example, in the expression 5t² + 7t, you will first identify the common factor in the given terms. Both terms have the variable t, but they do not share any common numerical factors. Thus, the GCF is simply t. When you begin factoring, you use the GCF to divide each term of the expression, which results in a simpler and more streamlined expression.

Algebraic Factorization

The process of algebraic factorization breaks down an algebraic expression into the product of simpler expressions, which can no longer be factored. Factorization is the reverse of expanding and is quite useful in solving equations and simplifying algebraic fractions. There are different methods of factorization, but finding the GCF is often the first step.

In the expression 5t² + 7t, after finding that the GCF is t, we factor it out, which means we divide both terms by t and write the expression as the product of the GCF and the remaining terms. The resulting factored form is t(5t + 7). This makes the structure of the expression more apparent and is important for further algebraic manipulation.

Polynomials

Polynomials are expressions that consist of variables and coefficients, structured as a sum of terms raised to non-negative integer powers. A polynomial can have constants, variables, and the exponents of variables, which should be whole numbers. Polynomials are classified by the number of terms they have, with special names like 'binomial' for two terms, and by their degree, which is the highest exponent among its terms.

Looking at our example, 5t² + 7t is a binomial polynomial — it has two terms. It's also called a second-degree polynomial because the highest exponent on the variable t is 2. Understanding polynomials is essential in algebra as they form the foundational blocks for more complex algebraic structures and are used in a wide array of problem-solving situations.