Problem 14 - Intermediate Algebra

Question

Is the following statement true or false? Justify your answer. $$ 2 x^{3}-9=2 x^{3}+0 x^{2}+0 x-9 $$

Step-by-Step Solution

Answer

The statement is true because both sides of the equation are identical.

Step 1: Understand the Equation

We have the equation \(2x^3 - 9 = 2x^3 + 0x^2 + 0x - 9\). We need to determine if this statement is true by checking if both sides of the equation are equal.

Step 2: Simplify Both Sides

Observe that both sides of the equation can be simplified. The left side is \(2x^3 - 9\), which is already simplified. The right side is \(2x^3 + 0x^2 + 0x - 9\), which simplifies to \(2x^3 - 9\) because adding terms with coefficients of zero doesn't change their values.

Step 3: Compare Both Sides

After simplification, both sides of the equation are \(2x^3 - 9\). Since both expressions are identical, the statement is true.

Key Concepts

Polynomial Equations

Polynomial equations involve expressions composed of variables and coefficients, structured around powers or exponents of the variable. In the given problem, we have the polynomial expression: \(2x^3 - 9 = 2x^3 + 0x^2 + 0x - 9\). Here, the highest power of the variable \(x\) is 3, making it a cubic polynomial.

To identify and compare polynomial equations, it's important to understand the parts they consist of:

• Terms: Each part of a polynomial is called a term, separated by addition or subtraction signs.
• Coefficients: These are the numbers in front of the variables. They can be positive, negative, or zero.
• Exponents: These indicate the power to which the variable is raised.

Recognizing these elements is essential in examining if two polynomial expressions are equivalent.

Simplification

Simplification in algebra is the process of combining like terms to create a more straightforward expression. In the equation from the original exercise, we simplify the right-hand side: \(2x^3 + 0x^2 + 0x - 9\).

Here's how simplification works:

• Combine like terms: Terms that have the same variable raised to the same power are combined. For example, \(0x^2\) and \(0x\) are like terms. Since both are multiplied by zero, they add nothing to the expression.
• Ignore zero terms: Any term with a coefficient of zero can be dropped, simplifying the equation to just \(2x^3 - 9\).

Understanding simplification helps keep equations manageable and reveals the true structure of the polynomials.

Equality in Algebra

Equality in algebra means that two expressions represent the same value, as explored in the exercise through \(2x^3 - 9 = 2x^3 - 9\). Establishing equality involves ensuring both expressions are equivalent after simplification.

Verifying equality is conducted through steps such as:

• Simplifying both expressions: Bring both sides to their simplest forms to ensure clarity in comparison.
• Comparing sides: Now simply compare the two sides. If they match identically, they are indeed equal.

Recognizing equality is fundamental in solving equations and confirming solutions. It ensures accuracy in the relationships among different algebraic expressions.