Problem 4 - Elementary and Intermediate Algebra

Question

Consider the binomial \(5 x+4\) a. What is the square of its first term? b. What is twice the product of its two terms? c. What is the square of its second term?

Step-by-Step Solution

Answer

a. \(25x^2\); b. \(40x\); c. \(16\).

Step 1: Identify the First Term

The binomial given is \(5x + 4\). The first term of this binomial is \(5x\). Identify this term for further calculations.

Step 2: Square the First Term

To find the square of the first term \(5x\), apply the formula \((5x)^2 = 25x^2\).

Step 3: Identify Both Terms

The terms in the binomial \(5x + 4\) are \(5x\) and \(4\).

Step 4: Calculate the Product of Two Terms

The product of the two terms \(5x\) and \(4\) is \(5x imes 4 = 20x\).

Step 5: Double the Product of Two Terms

Twice the product of \(5x imes 4\) is \(2 imes 20x = 40x\).

Step 6: Square the Second Term

The second term is \(4\). The square of this term is \(4^2 = 16\).

Key Concepts

Term Squaring

In the world of binomials, squaring each term individually is a fundamental concept. For the binomial expression \(5x + 4\), you're dealing with two terms: \(5x\) and \(4\). The process of term squaring involves taking one of these terms and multiplying it by itself.

For example, to square the first term \(5x\), you perform the operation \((5x) \times (5x)\), following the rule:

• Multiply the coefficient (which is 5 in this case) by itself, resulting in 25.
• Multiply the variable \(x\) by itself, resulting in \(x^2\).

Therefore, the square of the first term \(5x\) is \(25x^2\). Similarly, for the second term \(4\), you simply multiply \(4 \times 4 = 16\). Even though the second term lacks a variable, the squaring principle remains consistent, leading to \(16\).

Product of Terms

To find the product of the terms in a binomial, you simply multiply the two distinct terms together. Using the binomial \(5x + 4\), our terms are \(5x\) and \(4\). The key is to use the distributive property of multiplication across addition.

Set up your multiplication as \(5x \times 4\). Handle each component separately:

• 5x represents 5 times x, so \(5 \times x = 5x\).
• Multiplying 5x by 4 means distributing the 4 across the multiplication: \(5 \times 4 = 20\).

Thus, the product of these two terms is \(20x\). This foundational step is crucial as it often leads into further calculations like finding twice the product of these terms.

Twice the Product

The concept of "twice the product" in a binomial expression involves doubling the multiplication result of the binomial's two terms. To achieve this, first determine the original product, which we previously calculated as \(20x\) from the terms \(5x\) and \(4\).

Next, applying the idea of 'twice', you simply multiply the product result by 2. Here's how you can think about it:

• Take the product \(20x\).
• Multiply this by 2 to effectively double it: \(2 \times 20x\).
• This results in \(40x\), which is our doubled product.

Understanding this operation is important for solving many algebraic problems, especially those involving expansion formulas like \((a+b)^2\).