Problem 87 - College Algebra

Question

Perform the indicated operation or operations. $$ (2 x+5)(2 x-5)\left(4 x^{2}+25\right) $$

Step-by-Step Solution

Answer

The simplified form of the given expression \((2x + 5)(2x - 5)(4x^2 + 25)\) is \(16x^4 - 625\)

Step 1: Recognize Special Cases

The first two terms of the given problem are in the form of the difference of squares, where \(a = 2x\) and \(b = 5\). Remember that the general formula for the difference of squares is \(a^2 - b^2 = (a + b)(a - b)\). Thus, \((2x + 5)(2x - 5)\) becomes \((2x)^2 - 5^2\) or \(4x^2 - 25\)

Step 2: Apply Distributive Property

Next, the result from the previous step, \(4x^2 - 25\), is multiplied with the remaining function \((4x^2 + 25)\) which is accomplished by using the distributive property of multiplication over addition that is, \(a(b + c) = ab + ac\). You should expand the equation as \(4x^2*(4x^2 + 25) - 25*(4x^2 + 25)\)

Step 3: Calculate Each Term Independently

Now calculate each term independently. Multiply \(4x^2\) with \(4x^2\) and \(25\) separately and then multiply \(-25\) with \(4x^2\) and \(25\) separately, which yields \(16x^4 + 100x^2 - 100x^2 - 625\)

Step 4: Combine Like Terms

Finally, combine the like terms \(100x^2 - 100x^2\) which results in \(16x^4 - 625\)

Key Concepts

Distributive Property

Understanding the distributive property is crucial when working with algebraic expressions. It's a rule that allows us to multiply a single term by each term within a parenthesis. For example, in the context of the provided exercise \( (2x + 5)(2x - 5)(4x^2 + 25) \), we first recognize a difference of squares which simplifies to \(4x^2 - 25\).

Now, the distributive property comes into play as we multiply \(4x^2 - 25\) by \(4x^2 + 25\). Essentially, we distribute \(4x^2\) across the \(4x^2 + 25\) and then do the same with \( -25\). This is akin to sharing or dispersing something evenly. In algebraic terms, we express it as \(a(b+c)=ab+ac\).

For students working through this process, think of it like handing out apples (the \(4x^2\) and \( -25\)) to people (the terms inside the \(4x^2 + 25\)). Everyone gets an apple, which is just like every term in the parenthesis getting multiplied by the term outside.

Multiplying Polynomials

Once you've got the hang of the distributive property, multiplying polynomials will start to feel more approachable. In the exercise, after applying the distributive property, we need to multiply the terms. Polynomials are expressions that include variables raised to whole number exponents and their coefficients. To multiply them, we perform the distributive property multiple times.

Let's take \(4x^2*(4x^2 + 25)-25*(4x^2 + 25)\), as derived from the exercise. You're effectively multiplying each term of one polynomial by every term of the other. It's important to work systematically to ensure you multiply every term and don't miss any combinations.

For a clear understanding, imagine you're filling in a grid where each box represents the product of a term from one polynomial with a term from the other. The sum of the contents of this grid gives you the multiplied polynomial. It's similar to the multiplication you learned in elementary school, just performed with variables included.

Combining Like Terms

After distributing and multiplying, you'll likely end up with a more complex expression that needs simplifying. That's where combining like terms comes in. Like terms are terms in an algebraic expression that have the same variables raised to the same power. They can be combined to simplify the expression.

In our exercise, after multiplying the polynomials, we end up with \(16x^4 + 100x^2 - 100x^2 - 625\). You'll notice that \(+100x^2\) and \( -100x^2\) are like terms because they both have the variable \(x\) squared. Combining them means adding or subtracting their coefficients. Here, adding \(100\) and \( -100\) cancels out the \(x^2\) terms, leaving us with \(16x^4 - 625\).

Imagine you're organizing a toolbox and putting screws with screws and nails with nails. Combining like terms is like organizing; it makes the expression neater and more manageable.