Problem 24 - College Algebra with Corequisite Support

Question

For the following exercises, expand the binomial \((4 x+5)^{2}\)

Step-by-Step Solution

Answer

\(16x^2 + 40x + 25\)

Step 1: Identify the Binomial Structure

The expression given is \((4x + 5)^2\). This is a binomial raised to a power of 2, which means we will apply the formula for the square of a binomial: \((a + b)^2 = a^2 + 2ab + b^2\), where \(a\) and \(b\) are the terms inside the binomial.

Step 2: Square the First Term

In the binomial \((4x + 5)\), the first term \(a\) is \(4x\). So, we need to calculate \((4x)^2\). This results in \(16x^2\), because \(4^2 = 16\) and \((x)^2 = x^2\).

Step 3: Calculate Twice the Product of the Two Terms

Next, for the term \(2ab\), we take the product of \(a = 4x\) and \(b = 5\), and then multiply by 2. So, \(2 \times 4x \times 5 = 40x\).

Step 4: Square the Second Term

The second term \(b\) is \(5\). We compute \((5)^2\), which is \(25\) as \(5^2 = 25\).

Step 5: Combine the Results from Previous Steps

Combine all the results from Steps 2, 3, and 4 to write the expanded form. It becomes \(16x^2 + 40x + 25\).

Key Concepts

Algebraic Expressions

Algebraic expressions are foundational elements in mathematics. They are combinations of numbers, variables, and operations such as addition, subtraction, multiplication, and division. In the expression \((4x + 5)^2\), the term inside the parentheses, \(4x + 5\), is an algebraic expression. This particular expression contains:

• Coefficients: The number '4' in front of the variable 'x' is a coefficient, indicating how many times 'x' is to be counted.
• Variables: 'x' is the variable, representing an unknown value that can change.
• Constants: The number '5' is a constant, as it does not change.

Algebraic expressions can be simplified and manipulated according to the rules of algebra to solve equations and to expand expressions like binomials. Understanding these basic components is crucial for performing tasks such as binomial expansion, which involves breaking down expressions into their individual algebraic parts.

Polynomial Expansion

Polynomial expansion involves expressing a polynomial in an expanded format. This process is especially important when dealing with binomial expressions, such as \((4x + 5)^2\). By expanding the polynomial, we make it easier to perform operations like addition or multiplication with other polynomials. Here’s a breakdown of polynomial expansion:

• Identify the structure: Recognize expressions such as \((a + b)^2\), which indicates the need for a specific expansion formula.
• Apply the formula: Using the binomial expansion formula \((a + b)^2 = a^2 + 2ab + b^2\), replace \(a\) with \(4x\) and \(b\) with 5.
• Perform the calculations: Calculate each term separately — for \(a^2\), \(2ab\), and \(b^2\). When done, these are \(16x^2\), \(40x\), and \(25\) respectively for this example.

This process results in breaking down the binomial into its simplified polynomial form, in this case, \(16x^2 + 40x + 25\). The expansion allows for further operations and integration with other algebraic expressions, providing flexibility in solving complex equations.

Square of a Binomial

The square of a binomial is a common operation in algebra where a binomial expression is multiplied by itself. The general formula to remember for squaring a binomial like \((a + b)\) is:\[(a + b)^2 = a^2 + 2ab + b^2\]This formula shows that when you square a binomial, you:

• Square the first term: Multiply the first term by itself, i.e., \((a)^2\).
• Double the product of both terms: Calculate \(2ab\), which provides the middle term.
• Square the second term: Multiply the second term by itself, i.e., \((b)^2\).

For the binomial \((4x + 5)\), you apply this process:- First, square \(4x\) to get \(16x^2\).- Then, take twice the product of \(4x\) and \(5\) to get \(40x\).- Finally, square \(5\) to get \(25\).When these results are combined, the expanded expression is \(16x^2 + 40x + 25\). Understanding how to correctly square a binomial is key in advancing further in algebra and calculus, providing a solid foundation in polynomial operations.