Problem 87 - Algebra and Trigonometry

Question

In Exercises 83–90, perform the indicated operation or operations. $$ (2 x+5)(2 x-5)\left(4 x^{2}+25\right) $$

Step-by-Step Solution

Answer

The result of the operation is \(16x^4 - 625\)

Step 1: Multiply the first two binomials

Apply the binomial multiplication \((a+b)(a-b) = a^2 - b^2\), substitute \(a = 2x\) and \(b = 5\) we get \((2x + 5)(2x - 5) = (2x)^2 - 5^2 = 4x^2 - 25\)

Step 2: Multiply the resulting polynomial with the third Poynomial

Multiply the resulting polynomial from the previous step with the third polynomial. This can be done by multiplying each term of the first polynomial by each term of the second polynomial and then adding the results. So, \((4x^2 - 25)(4x^2 + 25) = (4x^2)^2 - (25)^2 = 16x^4 - 625\)

Step 3: Final Result

The multiplication of the three binomials given in the problem is \(16x^4 - 625\)

Key Concepts

Binomial Multiplication

Binomial multiplication is a fundamental aspect of algebra that deals with the multiplication of two binomials. A binomial is simply an algebraic expression that contains two terms, for example, \((2x + 5)\) and \((2x - 5)\). When multiplying two binomials, you use a specific rule known as the "difference of squares formula."

This formula is given by \((a + b)(a - b) = a^2 - b^2\). In the provided exercise, the binomials are \((2x + 5)\) and \((2x - 5)\). Here, \(a = 2x\) and \(b = 5\). Applying the formula, you calculate:

• \((2x)^2 = 4x^2\)
• \(5^2 = 25\)
• Thus, \((2x + 5)(2x - 5) = 4x^2 - 25\)

This method simplifies the process and avoids multiple steps, making it efficient to handle binomial multiplication.

Algebraic Expressions

Algebraic expressions are combinations of constants, variables, and operations such as addition, subtraction, multiplication, and division. In the exercise, the expression involves binomials and a polynomial. We started with \((2x + 5)(2x - 5)(4x^2 + 25)\).

An algebraic expression like \(4x^2 - 25\) is formed through operations between variables and numbers, illustrating how these combinations create more complex expressions.

• This example shows how distinct expressions can be multiplied to form a larger expression.
• It highlights the relationships between different parts of an expression and manipulations needed to simplify or solve them.

Understanding how to work with algebraic expressions is key to navigating complex mathematical problems, as it involves recognizing patterns and applying correct formulas and operations.

Distributive Property

The distributive property is a key principle in algebra that allows you to multiply a single term across terms inside a parenthesis. It states that for any numbers \(a\), \(b\), and \(c\), the equation \(a(b + c) = ab + ac\) holds true. In polynomial multiplication, the distributive property is used to simplify expressions.

When multiplying \((4x^2 - 25)(4x^2 + 25)\), each term of one polynomial must be distributed over every term of the other polynomial. This technique ensures that all parts of the polynomials are multiplied appropriately to get the final result.

• Here, you multiply each term in \(4x^2 - 25\) by each term in \(4x^2 + 25\).
• By doing so, the equation \((4x^2)^2 - (25)^2 = 16x^4 - 625\) is derived.

Mastering the distributive property is crucial as it simplifies complex problems and allows complete multiplication across multiple terms or expressions.