Problem 10 - Elementary Algebra

Question

State the restrictions and simplify. $$ 3 x 2+9 x 2 x 3-18 x $$

Step-by-Step Solution

Answer

The simplified expression is \(3x(19x - 6)\), and there are no restrictions.

Step 1: Identify Each Term

The given expression is \(3x^2 + 9x \cdot 2x \cdot 3 - 18x\). We first identify the different terms of the polynomial.1. \(3x^2\) is the first term.2. \(9x \cdot 2x \cdot 3\) is the second term, which we will simplify.3. \(-18x\) is the third term.

Step 2: Simplify Each Term

We simplify the term \(9x \cdot 2x \cdot 3\).Calculate the product: \[9x \cdot 2x \cdot 3 = (9 \cdot 2 \cdot 3)x^2 = 54x^2\]Now we have the expression as \(3x^2 + 54x^2 - 18x\).

Step 3: Combine Like Terms

Combine the like terms in the expression:- Combine \(3x^2\) and \(54x^2\): - \(3x^2 + 54x^2 = 57x^2\)This gives us the simplified expression: \(57x^2 - 18x\).

Step 4: Factor the Simplified Expression

Factor the expression \(57x^2 - 18x\) by taking the greatest common factor out:- The greatest common factor is \(3x\).- Factor it out: \[57x^2 - 18x = 3x (19x - 6)\]

Step 5: State Restrictions

Identify where the expression is undefined.Since this expression involves variables without denominators or even roots, the only restriction is when \(x = 0\) since the factored form involves \(x\), which cannot be zero in division. However, there is no division, so there are essentially no restrictions in this context.

Key Concepts

Understanding Like Terms

Like terms are terms in an algebraic expression that have the same variable raised to the same power. This means that the coefficients can be different, but the variable parts must be identical.

For example, in the expression \(3x^2 + 54x^2 - 18x\), the terms \(3x^2\) and \(54x^2\) are like terms because both contain \(x^2\). On the other hand, \(-18x\) is not a like term with the others because it contains only \(x\) instead of \(x^2\).

Identifying like terms is crucial as it allows you to combine them by simply adding or subtracting their coefficients.

• \(3x^2 + 54x^2 = 57x^2\)

This simplification makes the expression easier to work with.

Factoring Polynomials

Factoring polynomials involves rewriting the expression as a product of simpler expressions. This process makes it easier to solve equations or perform other algebraic operations.

In our example, after combining like terms, we had the expression \(57x^2 - 18x\). We then looked for the common factor to factor it.
To factor a polynomial:

• Identify the greatest common factor (GCF).
• Divide each term by the GCF.
• Express the polynomial as a product of the GCF and the resulting expression.

For \(57x^2 - 18x\), the GCF is \(3x\), and the expression can be factored to \(3x(19x - 6)\).

Finding Common Factors

Common factors are numbers or variables that divide exactly into each term of an algebraic expression. They are essential for simplifying expressions and solving equations.

In \(57x^2 - 18x\), we needed to find the common factor. Here's how to do it:

• List the factors of the coefficients (e.g., factors of 57 and 18).
• Identify the common variables in each term (e.g., \(x\)).

We found that 3 was the largest number that divided both 57 and 18. Since \(x\) is common in both terms, the GCF became \(3x\). This allows us to factor the expression to \(3x(19x - 6)\).

Finding common factors is a key step in polynomial simplification.

Exploring Algebraic Expressions

Algebraic expressions consist of terms that include numbers, variables, and operations. They represent a quantity or relationship and can vary in complexity.

Our expression \(3x^2 + 54x^2 - 18x\) is an example of a polynomial, which is a type of algebraic expression.
Characteristics of algebraic expressions:

• Contain one or more terms.
• Include operations such as addition, subtraction, multiplication.

To simplify algebraic expressions, we combine like terms and factor when possible. Simplifying makes expressions easier to understand and use for solving problems.

This process is fundamental in algebra and prepares you for more advanced concepts.