Problem 24 - Elementary Algebra

Question

Factor the greatest common factor from each polynomial. \(7 r+7\)

Step-by-Step Solution

Answer

7(r + 1)

Step 1: Identify the Common Factor

Examine each term in the polynomial to determine the greatest common factor (GCF). The terms are \(7r\) and \(7\). The GCF of these terms is \(7\).

Step 2: Factor Out the GCF

Express the polynomial as a product of the GCF and the remaining factors. Factor out \(7\) from each term: \(7(r + 1)\).

Step 3: Write the Final Factored Form

Combine the GCF and the remaining polynomial to write the final factored form: \(7(r + 1)\).

Key Concepts

greatest common factor

Understanding the greatest common factor (GCF) is crucial in algebra. The GCF of two or more numbers or terms is the largest factor that all terms share. Finding the GCF helps simplify polynomial expressions.

To identify the GCF of polynomial terms, follow these steps:

• List the factors of each term. For instance, if you have the terms 7r and 7, list the factors of each. For 7r, the factors are 1, 7, r, and 7r. For 7, the factors are 1 and 7.
• Identify the largest number that is a factor of all terms. Here, the largest common factor is 7.

Understanding how to find and use the GCF simplifies many algebraic operations and is essential when factoring polynomials.

factoring steps

Properly factoring polynomials involves a series of systematic steps. Applying these steps will ensure accurate results and a deeper understanding of algebra.

Here are the steps to follow when factoring polynomials:

• Step 1: Identify the Common Factor: Examine all terms in the polynomial to find the greatest common factor (GCF). For example, in the polynomial 7r + 7, the GCF is 7.
• Step 2: Factor Out the GCF: Once the GCF is found, divide each term in the polynomial by this factor. In 7r + 7, factor out 7 to get 7(r + 1).
• Step 3: Write the Final Factored Form: Express the polynomial as the product of the GCF and the remaining polynomial. The factored form of 7r + 7 is 7(r + 1).

By diligently following these steps, students can effectively simplify and factor polynomial expressions.

polynomial expressions

Polynomial expressions consist of terms that include variables raised to positive integer powers and coefficients. They appear frequently in algebra and higher-level mathematics.

Here are key points to understand about polynomial expressions:

• Terms: Each polynomial is made up of terms, which are products of coefficients and variables. In the expression 7r + 7, '7r' and '7' are terms.
• Degree: The degree of a polynomial is the highest power of the variable in its terms. For instance, the expression 2x^3 + 4x has a degree of 3.
• Simplification: Simplifying polynomials often involves combining like terms and factoring. Factoring a polynomial by its GCF, like turning 7r + 7 into 7(r + 1), simplifies the expression and reveals its structure.

Understanding these elements and how they interact is essential for mastering polynomials and succeeding in algebra.