Question

For Exercises, find the GCF of the monomials. \(8 x^{2}\) and \(14 x^{5}\)

Short answer

Answer: The GCF of the monomials \(8x^2\) and \(14x^5\) is \(2x^2\).

Step-by-step solution

Find the GCF of the coefficients

Find the GCF of the numerical part of the monomials, which are 8 and 14. In this case, the GCF is 2.

Find the lowest power of the variable

Examine the powers of \(x\) in both monomials. We have \(x^2\) and \(x^5\). The lowest power of \(x\) is \(x^2\).

Combine the results

Multiply the results of Step 1 and Step 2 to find the GCF of the given monomials. So, the GCF is \(2x^2\).

Key concepts

Greatest Common Factor

Understanding the Greatest Common Factor (GCF) is essential when simplifying mathematical expressions or solving problems involving polynomials. In essence, the GCF of two or more terms is the largest number (or expression, in the case of polynomials) that divides each of the terms without leaving a remainder. When finding the GCF of monomials, the process begins with identifying the GCF of the numerical coefficients. For example, with monomials like 8x^2 and 14x^5, we would first consider the constants 8 and 14 and determine that their GCF is 2.

Next, we look at the variables and their exponents. The key to getting the GCF of the variable part is to select the variable with the lowest exponent that occurs in all terms. Since we're dealing with exponents of the same base, this means we would choose x^2, as 2 is the smaller exponent in the terms x^2 and x^5. Finally, we combine the GCFs of the numerical and variable portions to obtain the overall GCF, which, in our example, is 2x^2. Hence, understanding GCF is crucial for simplification and comparative analysis of algebraic terms.

Polynomials

Polynomials are mathematical expressions consisting of variables and coefficients, combined using addition, subtraction, multiplication, and non-negative integer exponents of variables. A monomial, like 8x^2 or 14x^5, is a polynomial with a single term. When we talk about finding the GCF of monomials, we are seeking the largest polynomial that divides each monomial exactly.

It's important to understand that in the context of polynomials, the GCF may include literal (variable) parts along with the numerical coefficient. The process of finding the GCF is a foundational skill, as it streamlines various computations and simplifies expressions, which can be especially helpful when adding, subtracting, or factoring polynomials. By mastering the GCF, students can handle more complex polynomial operations with greater ease.

Exponents

Exponents play a pivotal role in algebra, particularly with polynomials. An exponent represents the number of times a base is multiplied by itself. For instance, in the expression x^2, the base is x and the exponent is 2, which means x is multiplied by itself once (x*x). When determining the common factors of monomials involving exponents, it is the lowest exponent that common bases across the terms share that becomes part of the GCF.

Understanding exponents is crucial for identifying the GCF in expressions that have variables raised to a power. In our case with x^2 and x^5, since the smaller exponent is 2, it becomes part of the GCF. Students should remember that when dealing with exponents in GCF calculations, we always choose the smallest exponent shared among the monomials for a given base. This ensures that the GCF is a factor of each term in the expression.