Problem 2 - Elementary Algebra

Question

If possible, factor the following binomials completely. $$ 36 p^{2}-81 q^{2} $$

Step-by-Step Solution

Answer

Answer: The completely factored form of the given binomial is 9(2p - 3q)(2p + 3q).

Step 1: Rewrite the binomial as a difference of squares

We need to recognize that the given binomial \((36p^2 - 81q^2)\) is a difference of squares. We can rewrite this as: $$(6p)^2 - (9q)^2$$

Step 2: Identify the square roots

Identify the square roots of each term. The square root of \((6p)^2\) is \(6p\), and the square root of \((9q)^2\) is \(9q\).

Step 3: Use the difference of squares formula

Apply the difference of squares formula: \((a^2 - b^2) = (a - b)(a + b)\). In this case, \(a = 6p\) and \(b = 9q\). So we have: $$(6p - 9q)(6p + 9q)$$

Step 4: Simplify (if possible)

In our factored form, we can see if there are any common factors between the coefficients and simplify further: $$(6p - 9q)(6p + 9q) = (3(2p) - 3(3q))(3(2p) + 3(3q))$$ Now, factor out the common factor of 3 from each term: $$3(2p - 3q)3(2p + 3q)$$ Finally, rewrite the expression as a product of two binomials: $$9(2p - 3q)(2p + 3q)$$ So, the completely factored form of the given binomial is \(9(2p - 3q)(2p + 3q)\).

Key Concepts

Difference of Squares

When working with algebraic expressions, recognizing patterns is essential. One common pattern is the "difference of squares." This term refers to an expression that is the result of subtracting one squared number from another. For example, in the expression given, \(36p^2 - 81q^2\), we have two perfect square terms: \((6p)^2\) and \((9q)^2\). This is where the difference of squares formula comes into play:

• Formula: \((a^2 - b^2) = (a - b)(a + b)\)

To apply this formula, identify each term being squared, then express the binomial as a product of two binomials: one adding and one subtracting these square roots. This pattern simplifies the factorization process significantly and is a powerful tool in solving algebraic equations.

Binomials

In algebra, understanding binomials is foundational for working with polynomials. A binomial is a sum or difference of two terms. In our exercise, \(36p^2 - 81q^2\) forms a binomial, being an expression with two distinct square terms.

• Two terms: Here, we see \(36p^2\) and \(81q^2\).
• Factorization: This specific expression's factorization was triggered by recognizing it as a difference of squares.

By factoring a binomial, especially as a difference of squares, one simplifies the problem into more manageable parts, making the expression easier to work with in broader equations or mathematical contexts.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations. They form the backbone of algebra. The expression we considered, \(36p^2 - 81q^2\), is a type of algebraic expression that involves subtraction of two squares. Key characteristics to understand include:

• Components: Consist of terms (here, squared terms), constants, and operators.
• Simplification: Through factorization methods like the difference of squares, these can be broken down into simpler expressions.

Recognizing kinds of expressions and mastering simplification techniques helps in algebra as it reduces complex expressions into simpler, more easily solvable forms and supports solving equations efficiently.