Problem 38 - Elementary Algebra

Question

For the following problems, find the products. $$ \left(x-\frac{2}{3}\right)^{2} $$

Step-by-Step Solution

Answer

Answer: The square of the binomial \(x - \frac{2}{3}\) is \(x^2 - \frac{4}{3}x + \frac{4}{9}\).

Step 1: Write down the given binomial and square it

We are given the following binomial to find the product: $$ \left(x-\frac{2}{3}\right)^{2} $$

Step 2: Use the general formula for squaring a binomial and plug in the values

We will use the general formula for squaring a binomial: $$ (a+b)^{2} = a^{2} + 2ab + b^{2} $$ In our case, \(a = x\) and \(b = -\frac{2}{3}\). Let's plug these values into the formula: $$ \left(x-\frac{2}{3}\right)^{2} = (x)^{2} + 2(x)\left(-\frac{2}{3}\right) + \left(-\frac{2}{3}\right)^{2} $$

Step 3: Simplify each term and find the final answer

Now, we have to simplify each term to find the final answer: $$ \left(x-\frac{2}{3}\right)^{2} = x^{2} - \frac{4}{3}x + \frac{4}{9} $$ So, the product is: $$ \left(x-\frac{2}{3}\right)^{2} = x^{2} - \frac{4}{3}x + \frac{4}{9} $$

Key Concepts

Squaring Binomials

Squaring binomials is a fundamental concept in algebra that entails expanding a binomial expression raised to the power of two. To square a binomial means to multiply the binomial by itself. For example, if given an expression like \((x + y)^2\), this is the same as \((x + y)(x + y)\).

Using a formula to expand such expressions efficiently is a handy skill. The standard formula for squaring binomials is:

• \((a + b)^2 = a^2 + 2ab + b^2\)

This formula allows us to quickly expand any squared binomial without having to perform lengthy multiplications. It's important to carefully substitute each part of the binomial into the formula as demonstrated in our exercise.

In the exercise given \((x - \frac{2}{3})^2\), the formula is applied with \(a = x\) and \(b = -\frac{2}{3}\). Ensuring each step is followed correctly will lead to the correct expansion and simplification of the binomial.

Algebraic Expressions

Algebraic expressions are combinations of constants, variables, and algebraic operations like addition, subtraction, and multiplication. They form the core building blocks of algebra.

Different parts of an expression serve unique roles:

• **Constants**: Fixed numbers, such as \(-\frac{2}{3}\) in our binomial example.
• **Variables**: Symbols that represent numbers, such as \(x\).
• **Coefficients**: Numbers multiplied by variables. For example, \(-\frac{4}{3}\) is the coefficient of \(x\) in the expanded form in the exercise.

Understanding each component is vital for manipulating and simplifying expressions.

Expressions can be simplified or expanded, as seen when squaring binomials. The goal is often to present an expression in its simplest form, which involves combining like terms and performing any possible arithmetic.

Polynomial Multiplication

Polynomial multiplication is the process of expanding expressions that contain two or more terms. A binomial is a simple polynomial with two terms, and multiplying binomials is a common task in algebra.

When multiplying polynomials, each term in one polynomial must be multiplied by each term in the other polynomial. For binomials like \((x - \frac{2}{3})\), squared, the multiplication follows the distributive or FOIL method:

• **First**: Multiply the first terms from each binomial, here \(x \times x = x^2\).
• **Outer**: Multiply the outer terms, \(x \times -\frac{2}{3} = -\frac{2}{3}x\).
• **Inner**: Multiply the inner terms, again \(-\frac{2}{3} \times x = -\frac{2}{3}x\).
• **Last**: Multiply the last terms, \(-\frac{2}{3} \times -\frac{2}{3} = \frac{4}{9}\).

Combining these results, especially the middle terms, yields the expanded form: \(x^2 - \frac{4}{3}x + \frac{4}{9}\).

Polynomial multiplication is crucial for transforming expressions and solving equations, forming a fundamental aspect of algebra.