Problem 24 - College Algebra

Question

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

Step-by-Step Solution

Answer

The expanded form is \(16x^2 + 40x + 25\).

Step 1: Understand the Formula

To expand the binomial \((4x + 5)^2\), we use the square of a binomial formula: \((a + b)^2 = a^2 + 2ab + b^2\). Here, \(a = 4x\) and \(b = 5\).

Step 2: Apply the Squaring Formula

Substitute \(a = 4x\) and \(b = 5\) into the formula: 1. Compute \(a^2\): \((4x)^2 = 16x^2\).2. Compute \(2ab\): \(2(4x)(5) = 40x\).3. Compute \(b^2\): \(5^2 = 25\).

Step 3: Add the Components

Combine all the parts: \[16x^2 + 40x + 25\]This is the expanded form of \((4x + 5)^2\).

Key Concepts

Square of a Binomial

When expanding a binomial, specifically squaring it, we employ a fundamental algebraic technique. The square of a binomial formula is \( (a + b)^2 = a^2 + 2ab + b^2 \). This formula is useful for converting the binomial into a simpler polynomial form. In the case of \( (4x + 5)^2 \), we assign \( a = 4x \) and \( b = 5 \). By following the formula, we calculate:

• First Term: \( a^2 = (4x)^2 = 16x^2 \)
• Middle Term: \( 2ab = 2 \times 4x \times 5 = 40x \)
• Last Term: \( b^2 = 5^2 = 25 \)

Bringing all these components together gives us the expanded expression \( 16x^2 + 40x + 25 \).
Understanding how to square binomials lays the foundation for working efficiently with more complex polynomial expressions.

Algebraic Expressions

Algebraic expressions consist of numbers, variables, and operations combined in a meaningful way. They allow us to represent real-world problems in mathematical terms.
For the expression \( 4x + 5 \), each part plays a significant role:

• The "\( 4x \)" is a term where "\( 4 \)" is the coefficient and "\( x \)" is the variable.
• The "\( 5 \)" is a constant term, meaning its value remains unchanged.

Operations involving algebraic expressions include addition, subtraction, multiplication, and squaring, as seen in our example.
By manipulating expressions using these operations, we solve equations and simplify expressions to find answers to algebraic problems.

Polynomial Expansion

The process of polynomial expansion involves using algebraic methods to simplify expressions. It converts a product of sums into individual terms, making them easier to handle. The square of a binomial is a specific case of polynomial expansion.
In \( (4x + 5)^2 \), we apply the binomial theorem, converting the repeated multiplication into a polynomial. By using the formula \( (a + b)^2 = a^2 + 2ab + b^2 \), we expand the binomial to a polynomial: \( 16x^2 + 40x + 25 \).
This expansion results in a quadratic polynomial, characterized by the highest exponent of the variable "\( x \)" being two. Each term in the polynomial provides insight into various properties of the equation, like its roots and graph.
Mastering polynomial expansion aids in simplifying complex algebraic expressions, solving equations, and understanding mathematical relationships.