Problem 64 - College Algebra with Modeling and Visualization

Question

Multiply the binomials. $$\left(x^{2}-2\right)\left(x^{2}+4\right)$$

Step-by-Step Solution

Answer

\(x^4 + 2x^2 - 8\)

Step 1: Distribute the First Term

Begin by distributing the first term of the first binomial, which is \(x^2\), to each term in the second binomial \((x^2 + 4)\). This gives us: 1. \(x^2 \cdot x^2 = x^4\)2. \(x^2 \cdot 4 = 4x^2\)

Step 2: Distribute the Second Term

Now distribute the second term of the first binomial, which is \(-2\), to each term in the second binomial \((x^2 + 4)\). This gives us: 1. \(-2 \cdot x^2 = -2x^2\)2. \(-2 \cdot 4 = -8\)

Step 3: Combine Like Terms

Now, combine all the terms obtained from the previous distributions:1. \(x^4 + 4x^2\) from Step 12. \(-2x^2 - 8\) from Step 2Combine like terms:- The \(x^2\) terms: \(4x^2 - 2x^2 = 2x^2\)The expression becomes: \(x^4 + 2x^2 - 8\).

Key Concepts

Algebraic Expressions

Algebraic expressions are a combination of variables, numbers, and operators like addition, subtraction, multiplication, and division. They form the basic building blocks of algebra and allow us to represent mathematical relationships. Each part of an algebraic expression separated by a plus or minus sign is called a 'term'. Terms can consist of constants (numbers on their own), variables (letters representing unknown or changeable values), or the product of both—known as coefficients when they multiply variables.
For example, in the expression \(3x^2 + 5x - 7\), there are three terms: \(3x^2\), \(5x\), and \(-7\). Each term can be identified by:

• Constant term: \(-7\)
• Linear term: \(5x\), which includes a variable raised to the power of 1
• Quadratic term: \(3x^2\), with the variable squared

Understanding how to work with algebraic expressions is crucial for solving equations and performing more complex operations, such as adding, subtracting, and multiplying polynomials.

Polynomial Operations

Polynomial operations involve various actions such as addition, subtraction, and multiplication of polynomials. A polynomial is a type of algebraic expression that includes more than one term. These terms are arranged in descending order of their degree, which is the highest power of the variable in the term.
When performing polynomial operations, like in the multiplication of the binomials \((x^2 - 2)(x^2 + 4)\), we apply certain rules to ensure the operations are carried out correctly.
To multiply polynomials, each term in one polynomial must be multiplied by every term in the other polynomial. This process results in a new polynomial. After multiplication, it's essential to combine like terms to simplify the expression. Like terms are those that contain the same variable raised to the same power.

• Addition: Sum the coefficients of like terms.
• Subtraction: Subtract the coefficients of like terms.
• Multiplication: Use the distributive property to distribute terms across the entire polynomial.

Mastering these operations enables us to handle more complex algebraic problems efficiently.

Distributive Property

The distributive property is a fundamental principle in algebra used to multiply a single term and two or more terms inside a parenthesis. It states that when you multiply a term by a sum, you can distribute the multiplication across each term inside the parenthesis. The formula for the distributive property is given by \( a(b + c) = ab + ac \).
This property simplifies expressions and equations, making it easier to solve them. Suppose you have the binomials \((x^2 - 2)(x^2 + 4)\). To multiply them, you would apply the distributive property:

• First, multiply \( x^2 \) by each term in \((x^2 + 4)\) to get: \(x^4 + 4x^2\).
• Then, multiply \(-2\) by each term in \((x^2 + 4)\) to get: \(-2x^2 - 8\).

Finally, combine the results to form the complete expression \(x^4 + 2x^2 - 8\). This step-by-step use of the distributive property ensures all terms are correctly multiplied and simplifies finding the final solution.