Problem 67 - Algebra and Trigonometry with Analytic Geometry

Question

Factor the polynomial. $$ 36 r^{2}-25 t^{2} $$

Step-by-Step Solution

Answer

The polynomial factors as \((6r - 5t)(6r + 5t)\).

Step 1: Identify Special Form

The given polynomial is in the form \(a^2 - b^2\) where \(a = 6r\) and \(b = 5t\). This is recognizable as a difference of squares.

Step 2: Apply Difference of Squares Formula

Recall that the difference of squares formula is \(a^2 - b^2 = (a - b)(a + b)\). Applying this to \(36r^2 - 25t^2\) gives us \((6r - 5t)(6r + 5t)\).

Step 3: Verify the Factorization

Expand \((6r - 5t)(6r + 5t)\) to ensure correctness: \((6r)(6r) + (6r)(5t) - (5t)(6r) - (5t)(5t) = 36r^2 - 25t^2\). The original polynomial is recovered, confirming the factorization is correct.

Key Concepts

Difference of Squares

The concept of "difference of squares" refers to an algebraic expression where two squares are subtracted from each other. It takes the form \( a^2 - b^2 \), where both \( a^2 \) and \( b^2 \) are perfect squares. This expression can be factored using the formula \( a^2 - b^2 = (a - b)(a + b) \).
This is a very useful identity in algebra because it simplifies the process of factoring certain polynomials.
In the provided example, the polynomial \( 36r^2 - 25t^2 \) fits the difference of squares format with \( a = 6r \) and \( b = 5t \). Recognizing this pattern helps to effortlessly apply the formula and find the factors.

Factoring Polynomials

Factoring polynomials is a core skill in algebra that involves expressing a polynomial as a product of its factors.
The purpose is to simplify polynomial expressions and solve polynomial equations.
The process requires recognizing patterns and using specific techniques, such as the difference of squares, to rewrite the expression.
In the case of \( 36r^2 - 25t^2 \), after identifying it as a difference of squares, we apply the formula \( (6r - 5t)(6r + 5t) \). This gives us the factors of the polynomial.

• Step 1: Recognize the form \( a^2 - b^2 \)
• Step 2: Write it as \( (a - b)(a + b) \)

This method is efficient and widely used for simplifying polynomial expressions.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations. They form the building blocks of algebra.
They vary in complexity from simple to complex polynomials. Understanding how to manipulate these expressions is key to mastering algebra.
They can be simplified, expanded, and factored to solve equations and understand mathematical relationships.

• It involves recognizing different forms such as linear, quadratic, and polynomial forms.
• This allows us to apply various techniques like factoring, expanding, and reducing expressions.

In the example \( 36r^2 - 25t^2 \), the expression is simplified by identifying it as a difference of squares and applying the factoring technique.

Quadratic Expressions

Quadratic expressions are polynomial expressions where the highest degree of any term is two. They typically take the form \( ax^2 + bx + c \).
When dealing with quadratic expressions, factoring is a common method for solving and simplifying them.
Although the given expression \( 36r^2 - 25t^2 \) doesn't have a middle term \( bx \), it is structurally similar to a quadratic because of the squared terms. Recognizing such expressions and using the appropriate factoring methods, such as the difference of squares, is crucial.Understanding quadratic expressions allows one to identify roots and solutions effectively, aiding in various mathematical applications.