Problem 47 - Algebra and Trigonometry

Question

In Exercises 15–58, find each product. $$ \left(4 x^{2}-1\right)^{2} $$

Step-by-Step Solution

Answer

The square of the given expression \( (4x^2 - 1) \) is \( 16x^4 - 8x^2 + 1 \)

Step 1: Write down the expression to be squared

The given expression is \( (4x^2 - 1) \) and we are required to find the square of it. This means we would multiply this expression by itself i.e. \( (4x^2 - 1) \times (4x^2 - 1) \)

Step 2: Apply the FOIL method

We now apply the FOIL method:\nFirst terms: \( (4x^2) \times (4x^2) = 16x^4 \),\nOuter terms: \( (4x^2) \times (-1) = -4x^2 \), \nInner terms: \( (-1) \times (4x^2) = -4x^2 \), \nLast terms: \( (-1) \times (-1) = 1 \).

Step 3: Sum up all the results

Summing up all the terms from the previous steps, we get \( 16x^4 - 4x^2 - 4x^2 + 1 = 16x^4 - 8x^2 + 1 \)

Key Concepts

FOIL Method

When multiplying two binomials, the FOIL method is a handy tool to use. FOIL is an acronym that stands for First, Outer, Inner, and Last, which are the specific terms you multiply together from each binomial. This method helps in systematically breaking down the multiplication process.

• First: Multiply the first terms in each binomial.
• Outer: Multiply the outer terms in the expression.
• Inner: Multiply the inner terms.
• Last: Multiply the last terms from each binomial.

After calculating each component, you sum them all up to get your final expression. In our given problem, using the FOIL method helped us expand \( (4x^2 - 1) imes (4x^2 - 1) \) into its components, eventually leading to the polynomial \( 16x^4 - 8x^2 + 1 \).
This technique is crucial in algebra as it sets the foundation for more complex polynomial multiplications.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and operations. Understanding their structure is essential for manipulating them through operations like addition, subtraction, and multiplication, as seen in polynomial work.
In the problem at hand, we deal with the expression \( (4x^2 - 1) \). This is a common structure, a binomial made up of a term with a variable, \( 4x^2 \), and a constant, \(-1\).
Key Points:

• Variables: Represented by letters, in this case, \( x \).
• Coefficients: Numbers multiplying the variables, here \( 4 \).
• Constants: Fixed values, such as \(-1\).

Recognizing these components is vital because it allows us to apply techniques, like FOIL, effectively. Being comfortable with algebraic expressions makes the expansion and simplification processes easier.

Squaring Binomials

Squaring binomials means multiplying the binomial by itself. This is a specific case of polynomial multiplication that simplifies certain processes in algebra. When you square a binomial, you're using pattern recognition to make the process more efficient.For example, squaring \( (4x^2 - 1) \) involves multiplying it by itself. This results in employing the formula:\[(a - b)^2 = a^2 - 2ab + b^2\].
Here, \( a = 4x^2 \) and \( b = 1 \). Using this squaring formula helps us directly arrive at the result \( 16x^4 - 8x^2 + 1 \).
Understanding this concept allows you to quickly expand and simplify expressions, saving you time and reducing errors.

Polynomial Expansion

Polynomial expansion involves expressing a product of polynomials as a sum of terms. The goal is to simplify or rearrange it into a more workable form.This process permits us to see all terms clearly and either combine like terms or make other algebraic manipulations as necessary.
In the context of squaring \((4x^2 - 1)\), through polynomial expansion, we used both FOIL for individual term multiplication and algebraic properties to sum up the results.The final product we obtained, \(16x^4 - 8x^2 + 1\), is a fully expanded polynomial form.
Studying polynomial expansion can greatly aid in simplifying more complex expressions and equations, opening up pathways to solve or graph them.