Problem 48 - Algebra 1: Concepts and Skills

Question

Tell whether the statement is true or false. If the statement is false, rewrite the right-hand side to make the statement true. $$ (3 s+2 t)(3 s-2 t)=9 s^{2}+4 t^{2} $$

Step-by-Step Solution

Answer

The given equation is false. The correct equation is \( (3s + 2t) (3s - 2t) = 9s^{2} - 4t^{2} \).

Step 1: Identifying the Problem

The problem statement provides an equation, that is \( (3s + 2t) (3s - 2t) = 9s^{2} + 4t^{2} \). The task is to determine whether this statement is true or false and to correct it if it is incorrect.

Step 2: Applying the Difference of Two Squares

When we have an equation in the form \( (a + b) (a - b) \) where \(a\) is \(3s\) and \(b\) is \(2t\), the result is \(a^{2} - b^{2}\). Therefore, expanding the left-hand side of the equation using this rule, we get \( (3s)^{2} - (2t)^{2} \) which simplifies to \( 9s^{2} - 4t^{2} \).

Step 3: Compare With Given Right-hand Side

We can now compare this result with the right-hand side of the given equation, which is \( 9s^{2} + 4t^{2} \). They are not equal because the sign in the middle term is different.

Step 4: Correcting the Equation

To correct the equation, replace the plus sign in the right-hand side of the original equation with a minus sign. Hence, the correct expression should be \( (3s + 2t) (3s - 2t) = 9s^{2} - 4t^{2} \).

Key Concepts

Algebraic Identities

Algebraic identities are equations that hold true for all values of the variables involved. They are the backbone of simplifying algebraic expressions and solving algebraic equations. One particularly useful identity is the difference of two squares, which states that the product of a sum and a difference of the same two terms equals the difference between the squares of those terms.

The general form of this identity is \[ (a + b)(a - b) = a^{2} - b^{2} \.\]
Using this identity, you can quickly factor or expand expressions without the need for long multiplication. This identity is especially handy when dealing with quadratic equations and can be a time-saver for complex polynomial expressions.

Factoring Polynomials

Factoring polynomials is a process of breaking down a complex expression into simpler components or 'factors' that, when multiplied together, produce the original polynomial. The difference of two squares is a classic factoring technique often utilized in factoring polynomials.

For example, when you come across a polynomial such as \( x^{2} - 9 \) which is a difference of two squares, it can be factored into \( (x + 3)(x - 3) \).

Importance of Factoring

Factoring is not just a methodical step; it's crucial for:

• Simplifying expressions to their lowest terms.
• Finding roots or solutions of polynomials.
• Dividing polynomials by splitting them into factors.
• Solving algebraic equations efficiently.

Being adept at factoring can significantly ease understanding and solving algebraic problems, thereby enhancing one's proficiency in mathematics.

Binomial Expressions

A binomial expression is an algebraic expression containing two terms, typically connected by a plus or minus sign. The difference of two squares exercise we've been discussing involves a pair of binomial expressions. Each binomial in this context takes the form of \( a + b \) and \( a - b \), where \( a \) and \( b \) are any algebraic expressions.

Understanding how to work with binomials is vital as they frequently occur in algebra. When multiplied, they can produce various types of results depending on their structure:

• Sum and Difference: \( (a + b)(a - b) = a^{2} - b^{2} \)
• Square of a Binomial: \( (a + b)^{2} = a^{2} + 2ab + b^{2} \)
• Product of Sums: \( (a + b)(c + d) = ac + ad + bc + bd \)

Recognizing these patterns and understanding how to apply the correct identity or operation with binomials is a key skill in algebra.