Problem 8 - Algebra 2 and Trigonometry

Question

In \(3-8,\) write each polynomial as the product of its greatest common monomial factor and a polynomial. $$ 21 a^{2}-14 a+7 $$

Step-by-Step Solution

Answer

The polynomial is \(7(3a^2 - 2a + 1)\).

Step 1: Identify the greatest common factor (GCF)

To factor the given polynomial, first identify the greatest common monomial factor of all the terms in the polynomial. The terms are \(21a^2\), \(-14a\), and \(7\). The coefficients are 21, 14, and 7, and the common factor here is 7. Therefore, the GCF is 7.

Step 2: Factor out the GCF from each term

Divide each term of the polynomial by the GCF, which is 7, to express the polynomial as a product. \(\frac{21a^2}{7} = 3a^2\), \(\frac{-14a}{7} = -2a\), and \(\frac{7}{7} = 1\).

Step 3: Write the polynomial as a product

Now express the original polynomial as the product of the GCF and the resultant polynomial from step 2. The polynomial \(21a^2 - 14a + 7\) can be written as the product \(7(3a^2 - 2a + 1)\).

Key Concepts

Greatest common factor (GCF)

The Greatest Common Factor (GCF) is a key concept in polynomial factorization. It involves finding the largest integer or expression that divides all terms of a polynomial, leaving no remainder. This shared factor helps simplify polynomial expressions by deconstructing them into simpler forms, making complex calculations a lot easier to handle.

In the exercise, the expression given is a polynomial consisting of terms: \(21a^2\), \(-14a\), and \(7\). By identifying the GCF, which in this case is \(7\), each term is divided by \(7\) to simplify the expression. The coefficients \(21\), \(-14\), and \(7\) all share \(7\) as a factor, which makes \(7\) the GCF. This process highlights the importance of recognizing common factors as an essential step in polynomial factorization. Understanding how to find the GCF effectively can help simplify various algebraic processes.

Monomials

Monomials are an essential building block of polynomials. A monomial is an algebraic expression that consists of a single term, which can either be a number, a variable, or a product of numbers and variables raised to non-negative integer powers. For instance, \( 21a^2 \) and \( 7\) are monomials.

When working with polynomials, recognizing each component monomial can help in identifying common factors. In our original exercise, each component of the polynomial \(21a^2 - 14a + 7\) is a monomial. This breakdown into simple building blocks allows for more straightforward operations, such as finding the GCF and factoring the expression. By perceiving a polynomial as a collection of monomials, you can apply factorization techniques systematically and efficiently.

Polynomial expression

A polynomial expression is a mathematical expression formed by summing monomials, which are known as terms. Polynomials are classified based on the number of terms and their degree. The degree of a polynomial is determined by the highest power of the variable present in the expression.

In the given exercise, the polynomial expression \(21a^2 - 14a + 7\) is a second-degree polynomial because the highest power of the variable \(a\) is 2 (in the term \(21a^2\)). This expression is a trinomial, consisting of three terms. To simplify or solve problems involving polynomials, one often looks for opportunities to factor the expression. By identifying common factors and understanding the structure of polynomials, algebraic manipulations become more manageable. This aids in operations like solving equations, graphing functions, and modeling real-world scenarios.