Problem 75 - Algebra A Combined Function

Question

Mixed Practice Multiply. $$ (4 x+5)(4 x-5) $$

Step-by-Step Solution

Answer

The expression simplifies to \(16x^2 - 25\).

Step 1: Identify the Formula

Recognize that the expression follows the pattern of a difference of squares. The general formula for a difference of squares is \((a + b)(a - b) = a^2 - b^2\). In this problem, we identify \(a = 4x\) and \(b = 5\).

Step 2: Apply the Formula

Substitute \(a = 4x\) and \(b = 5\) into the difference of squares formula: \[(4x + 5)(4x - 5) = (4x)^2 - 5^2\]

Step 3: Compute the Squares

Calculate \((4x)^2\) and \(5^2\): \[(4x)^2 = (4)^2(x)^2 = 16x^2\]\[5^2 = 25\]

Step 4: Simplify the Expression

Combine the results from Step 3 to express the simplified form: \[16x^2 - 25\]

Key Concepts

Polynomial Multiplication

Polynomial multiplication is a fundamental algebraic skill that involves distributing the terms from one polynomial across the other. In this specific problem, we are dealing with the multiplication of two binomials, specifically of the form \((a + b)(a - b)\).
Understanding this pattern is crucial because it is a special case known as the difference of squares.

• When multiplying polynomials, each term in the first polynomial is multiplied by each term in the second.
• For binomials, each term in the first binomial multiplies both terms in the second binomial, leading to four terms initially.
• However, with a difference of squares, the middle terms cancel out, simplifying the process immensely.

A good grasp of distributing terms aids in recognizing such patterns quickly, saving time and effort in simplifying expressions.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operators (like addition or multiplication). Identifying patterns in expressions, like recognizing a difference of squares, can greatly ease simplifying problems.
In this context:

• Variables like \(x\) represent unknown quantities and are manipulated using algebraic rules.
• Exponents indicate repeated multiplication of a variable by itself, such as \((4x)^2 = 16x^2\).
• The structure of the expression determines how you simplify it, as seen with the expression \( (4x + 5)(4x - 5) \) corresponding to a recognizable pattern.

By converting algebraic expressions into known patterns, complex multiplication can be simplified into familiar operations, making problems easier to handle.

Simplifying Expressions

Simplifying expressions involves reducing them to their simplest form, often by eliminating terms or combining like terms. In this problem, after applying the difference of squares theorem:
The expression \((4x+5)(4x-5)\) is reduced to \(16x^2 - 25\).
Here's how:

• Recognize the special form, which simplifies the multiplication with minimal calculation.
• Compute individual squares: \((4x)^2 = 16x^2\) and \(5^2 = 25\).
• Subtract the results as per the difference of squares formula \(a^2 - b^2\).

The final expression \(16x^2 - 25\) represents the simplest form, ensuring all calculations have been appropriately applied and checked. Simplifying expressions helps in not just reducing computational work but also in uncovering deeper mathematical insights.