Problem 20 - Elementary Algebra

Question

For the following problems, the first quantity represents the product and the second quantity a factor. Find the other factor. $$ 9 x^{2}+6 x+18,6 $$

Step-by-Step Solution

Answer

Answer: The other factor is \(\frac{1}{2}(3x^2 + 2x + 6)\).

Step 1: Examine the expression

Observe the given expression: $$ 9x^2 + 6x + 18 $$ It's a quadratic expression, which means it's a polynomial with a degree of 2.

Step 2: Factor the expression

To find the other factor, we must first factor the given expression. Find the greatest common divisor of the expression's terms. The greatest common divisor of \(9x^2\), \(6x\), and \(18\) is \(3\). Thus, we'll factor the expression by taking out the common divisor: $$ 9x^2 + 6x + 18 = 3(3x^2 + 2x + 6) $$

Step 3: Divide factored expression

We are given a factor of \(6\). Divide the factored expression by \(6\) to find the other factor: $$ \frac{3(3x^2 + 2x + 6)}{6} = \frac{1}{2}(3x^2 + 2x + 6) $$

Step 4: Write the final answer

The other factor is: $$ \frac{1}{2}(3x^2 + 2x + 6) $$

Key Concepts

Polynomial Factorization

Polynomial factorization is an essential skill in algebra that involves breaking down a polynomial into a product of smaller polynomials or factors. This process simplifies many types of algebraic problems, such as solving equations and simplifying expressions. For instance, the polynomial expression in the form of \( ax^2 + bx + c \), which describes a quadratic expression, can often be decomposed into two binomial factors. The goal is to find two binomials that when multiplied together, return the original quadratic expression.

The process starts by looking for common factors in all the terms of the polynomial, then applying various techniques like grouping, using the quadratic formula, or special factorization formulas like the difference of squares and sum and difference of cubes. After factorization, the expression becomes much easier to work with, especially for further algebraic operations like simplification or finding roots of equations.

Greatest Common Divisor

The greatest common divisor (GCD), also known as the greatest common factor (GCF), is the largest positive integer that divides two or more integers without leaving a remainder. In algebra, identifying the GCD of the coefficients in a polynomial is a critical step in factorization.

For example, in the algebraic expression \( 9x^2 + 6x + 18 \), the GCD of the coefficients, 9, 6, and 18, is 3. By dividing each term by 3, you can extract the common divisor as a factor. This extraction results in a simpler expression that holds the same value as the original. Mathematically, this process is represented as \( 3(3x^2 + 2x + 6) \). Understanding the concept of the GCD can greatly aid in simplifying expressions and solving algebraic equations efficiently.

Algebraic Expressions

Algebraic expressions are combinations of constants, variables, and arithmetic operations like addition, subtraction, multiplication, and division. In algebra, these expressions represent mathematical relationships and can take many forms such as monomials, binomials, trinomials, and higher degree polynomials.

The expression in our exercise, \( 9x^2 + 6x + 18 \), is a trinomial since it consists of three terms. Upon factoring, we aim to distill this expression into its fundamental components, thereby simplifying complex algebraic problems. Algebraic expressions are the backbone of algebra and effectively working with them is crucial. Through manipulation and transformation, like factoring, expanding, and simplifying, we solve for unknown variables, evaluate expressions, and explore the structure of algebraic relationships.