Question

$$ \text { perform the indicated operations and simplify. } $$ $$ (2 x-3)^{2} $$

Short answer

The simplified expression is \(4x^2 - 12x + 9\).

Step-by-step solution

Understand the Problem

We need to simplify the expression \((2x - 3)^2\). This means we have to expand it using algebraic operations.

Apply the Binomial Theorem

The expression \((a - b)^2\) is expanded using the formula \((a - b)^2 = a^2 - 2ab + b^2\). Here, our \(a\) is \(2x\) and \(b\) is \(3\).

Square the First Term

The first term in the formula is \(a^2\). So, we square \(2x\): \((2x)^2 = 4x^2\).

Calculate the Product of the Middle Term

The middle term in the formula is \(-2ab\). Multiply \(-2\), \(2x\), and \(3\): \(-2 \times 2x \times 3 = -12x\).

Square the Last Term

The last term in the formula is \(b^2\). So, we square \(3\): \(3^2 = 9\).

Combine the Terms

Now, we combine all the results from the previous steps to simplify the expression: \(4x^2 - 12x + 9\).

Key concepts

Algebraic Operations

Algebraic operations are foundational tools in mathematics that allow us to manipulate numbers and variables to solve equations and simplify expressions. These operations include addition, subtraction, multiplication, and division, which can be applied to both numbers and variables.

The key to mastering algebraic operations is understanding how to handle numbers and variables when they are part of complex expressions. For example, when performing binomial expansion, each term in the expression must be carefully calculated through multiplication and addition or subtraction of coefficients and variables.

• Addition and Subtraction: Combine like terms by adding or subtracting their coefficients.
• Multiplication: Distribute each term in one polynomial by every term in another.
• Division: When dividing polynomials, apply the long division or synthetic division method.

These operations are not just critical in solving equations, but also in simplifying expressions, essentially making complex problems easier to understand and work with.

Simplifying Expressions

Simplifying expressions involves reducing an algebraic expression to its simplest form while maintaining its original value. This process often requires the application of algebraic operations to combine like terms and eliminate unnecessary components.

In the provided exercise, we aimed to simplify the expression \( (2x - 3)^2 \). Using the binomial expansion formula, we expanded it to \( 4x^2 - 12x + 9 \). Here's how to simplify effectively:

• Identify like terms: These are terms that involve the same variable raised to the same power.
• Combine like terms: Add or subtract the coefficients of like terms to reduce the expression.
• Simplify constants: Basic arithmetic simplifies numbers.

By simplifying expressions, we not only make them easier to read but also easier to evaluate. This step is essential in solving real-world problems, as it helps streamline complex mathematical models.

Polynomials

Polynomials are algebraic expressions consisting of variables raised to various powers, multiplied by coefficients, and combined using addition or subtraction. They can range from simple expressions like \( x + 1 \) to more complex forms like \( 4x^2 - 12x + 9 \).

The degree of a polynomial is defined by the highest power of the variable present, which provides insights into the function's behavior. For example, in the expression resulting from our exercise, \( 4x^2 - 12x + 9 \, \) the degree is 2, indicating a quadratic polynomial.

• Understanding polynomial degrees helps predict the number of solutions or roots the expression may have.
• Polynomials can be added, subtracted, multiplied, or divided just like numbers.
• They are also easily differentiated and integrated in calculus.

Knowing how to work with polynomials is crucial for mathematical modeling and solving higher-level math problems. Whether in algebra or calculus, the principles governing polynomials form the backbone of many mathematical concepts and applications.