Problem 26 - Algebra A Combined Function

Question

Multiply. $$ \left(x^{2}+0.3\right)^{2} $$

Step-by-Step Solution

Answer

The expanded form is \(x^4 + 0.6x^2 + 0.09\).

Step 1: Apply the Binomial Expansion Formula

For any binomial \((a + b)^2\), the expansion formula is \(a^2 + 2ab + b^2\). Here, you can consider \(a = x^2\) and \(b = 0.3\). So substitute these into the formula.

Step 2: Calculate Term \(a^2\)

Substitute \(a = x^2\) into the term \(a^2\). This results in \((x^2)^2 = x^4\).

Step 3: Calculate the Middle Term \(2ab\)

Substitute \(a = x^2\) and \(b = 0.3\) into the term \(2ab\). This results in \(2 \cdot x^2 \cdot 0.3 = 0.6x^2\).

Step 4: Calculate Term \(b^2\)

Substitute \(b = 0.3\) into the term \(b^2\). This results in \((0.3)^2 = 0.09\).

Step 5: Combine the Terms

Combine the results from Steps 2, 3, and 4: \[x^4 + 0.6x^2 + 0.09\] is the expanded expression for \((x^2 + 0.3)^2\).

Key Concepts

Algebra

Algebra is a branch of mathematics that deals with symbols and the rules for manipulating those symbols. It is all about finding the unknowns in mathematical problems and creating equations to represent real-world scenarios in simplified forms. Algebra allows us to model situations and solve problems in a streamlined way.

In algebra, symbols such as letters are often used to represent numbers in expressions and equations. These symbols are known as variables. Through algebra, we can express relationships between these variables and solve for unknown values.

Consider the basic equation:

• \(x + 2 = 5\)

Here, \(x\) is the unknown, and the goal is to find its value by isolating it on one side of the equation. Understanding algebraic principles is fundamental in tackling more complex mathematical operations.

Polynomial Multiplication

Polynomial multiplication involves multiplying expressions that include variables raised to various powers. When dealing with polynomials, you must expand them by multiplying out the terms.

A polynomial is an expression containing multiple terms, connected by addition or subtraction. For instance, when multiplying the binomial

• \((x^2 + 0.3)\)

by itself, it's crucial to apply rules like the Binomial Expansion Formula. This formula simplifies the process and avoids errors when handling terms with multiple variables or exponents.

Exponents

Exponents represent repeated multiplication of a number or variable by itself. They are a compact way to describe a number multiplied by itself multiple times. For example, \(x^3\) represents \(x\times x\times x\).

In the context of our exercise, applying exponents correctly is essential when using the Binomial Expansion Formula. Here, you have terms like \(x^4\) which results from multiplying \((x^2)\) by itself.

Understanding how to manipulate exponents will help in simplifying and solving polynomial expressions efficiently, making it possible to work with complex algebraic operations.

Mathematical Expressions

Mathematical expressions are combinations of numbers, variables, and operators (such as +, -, *, and /) that signify a particular calculation. Expressions are different from equations, which equate two expressions.

In our given exercise, the expression \((x^2 + 0.3)^2\) is expanded into \(x^4 + 0.6x^2 + 0.09\), showcasing the transformation and simplification of the original expression. Mathematical expressions often need detailed analysis and a step-by-step process to compute accurately, as illustrated in this example.

It's crucial to be precise with each calculation to ensure that the expanded form of the expression retains the same value as the original.