Problem 68 - Algebra and Trigonometry with Analytic Geometry

Question

Factor the polynomial. $$ 81 r^{2}-16 t^{2} $$

Step-by-Step Solution

Answer

The polynomial factors as \((9r - 4t)(9r + 4t)\).

Step 1: Recognize the Form

The expression \(81r^{2} - 16t^{2}\) is a difference of squares. The difference of squares formula is \(a^{2} - b^{2} = (a-b)(a+b)\).

Step 2: Identify the Squares

Identify the squares in the expression. Here \(81r^{2} = (9r)^{2}\) and \(16t^{2} = (4t)^{2}\).

Step 3: Apply the Difference of Squares Formula

Apply the difference of squares formula. Replace \(a\) with \(9r\) and \(b\) with \(4t\) to factor the expression: \[81r^{2} - 16t^{2} = (9r - 4t)(9r + 4t)\]

Step 4: Verify the Factorization

Expand \((9r - 4t)(9r + 4t)\) to verify the factorization. Using the distributive property:\[(9r - 4t)(9r + 4t) = 9r \cdot 9r + 9r \cdot 4t - 4t \cdot 9r - 4t \cdot 4t\]Simplifying this, you get:\[81r^2 + 36rt - 36rt - 16t^2 = 81r^2 - 16t^2\]This confirms that the factorization is correct.

Key Concepts

Difference of Squares

The term "difference of squares" refers to a specific type of polynomial that can be factored in a unique way. A difference of squares has the form \(a^2 - b^2\). This expression can be factored into \((a-b)(a+b)\). The process of factoring polynomials using this formula is straightforward and highly useful in simplifying equations.
To apply it, identify if your polynomial fits the difference of squares pattern. If each term is a perfect square and they are subtracted, you are dealing with a difference of squares:

• First, determine the square root of each squared term.
• Express the polynomial in the form \(a^2 - b^2\).
• Then, rewrite it as \((a-b)(a+b)\).

For example, the polynomial \(81r^{2} - 16t^{2}\) can be rewritten as \((9r)^{2} - (4t)^{2}\), making \(a\) equal to \(9r\) and \(b\) equal to \(4t\). You apply the difference of squares formula to factor it into \((9r - 4t)(9r + 4t)\).

Algebraic Expressions

Algebraic expressions consist of variables, constants, and operations and are fundamental in mathematics. They can represent real-world problems that you can solve using algebraic manipulation. Understanding how to handle algebraic expressions, such as factoring, is crucial.When approaching algebraic expressions, remember:

• Terms: Parts of the expression separated by addition or subtraction.
• Coefficients: Numbers multiplied by the variables in each term.
• Constants: Numbers without variables.

To simplify or solve these, you may need to factor them using different techniques such as the distributive property or the difference of squares.
In our example, \(81r^{2} - 16t^{2}\) is an algebraic expression involving variables \(r\) and \(t\). By recognizing it as a difference of squares, you can simplify it further.

Distributive Property

The distributive property is a fundamental rule in algebra that allows you to simplify expressions and expand factored forms. It states that for any numbers or expressions \(a\), \(b\), and \(c\), you have: \(a(b+c) = ab + ac\).
This property is used frequently, especially when verifying factored expressions.\ While factoring \(81r^2 - 16t^2\), we check the factorization by expanding \((9r - 4t)(9r + 4t)\). Use the distributive property:

• First: Multiply \(9r\) by both \(9r\) and \(4t\).
• Next: Multiply \(-4t\) by both \(9r\) and \(4t\).
• Then: Combine like terms and simplify.

When expanded, these steps lead you back to the original expression. Thus, the distributive property confirms the correctness of the factorization.