Problem 52 - Algebra and Trigonometry

Question

31–76 ? Factor the expression completely. $$ 4 t^{2}-9 s^{2} $$

Step-by-Step Solution

Answer

The expression factors to \((2t - 3s)(2t + 3s)\).

Step 1: Recognize the Expression Type

The expression \(4t^2 - 9s^2\) is a subtraction of squares, which fits the form \(a^2 - b^2\). This is called a difference of squares, which can be factored using the formula \(a^2 - b^2 = (a - b)(a + b)\).

Step 2: Identify the Squares

Identify the squares in the expression: \(4t^2\) is \((2t)^2\) and \(9s^2\) is \((3s)^2\). This makes the expression \((2t)^2 - (3s)^2\).

Step 3: Apply the Difference of Squares Formula

Use the difference of squares formula: \((a - b)(a + b)\) for \((2t)^2 - (3s)^2\). Substitute \(a = 2t\) and \(b = 3s\) into the formula, resulting in \((2t - 3s)(2t + 3s)\).

Step 4: Verify the Factorization

Verify by multiplying the factors back: \((2t - 3s)(2t + 3s) = 4t^2 + 6ts - 6ts - 9s^2 = 4t^2 - 9s^2\). The middle terms, \(+6ts\) and \(-6ts\), cancel each other, confirming the correct factorization.

Key Concepts

Factoring

Factoring is a fundamental operation in algebra that involves breaking down an expression into simpler components, known as factors. It turns a complex expression into a product of simpler expressions.
For example, when you see a polynomial like

• \(4t^2 - 9s^2\)

you may notice it can be simplified. This is done by identifying parts of the expression that are perfect squares.
In this case,

• \(4t^2\) can be rewritten as \((2t)^2\)
• and \(9s^2\) as \((3s)^2\)

.
Recognizing these perfect squares is crucial in the factoring process, especially for the difference of squares, which allows us to use a specific formula to simplify the expression.

Algebraic Expressions

Understanding algebraic expressions is fundamental when learning algebra. An algebraic expression consists of numbers, variables, and arithmetic operations. It can be as simple as a single number or as complex as a multi-variable polynomial.
For instance, consider

• \(4t^2 - 9s^2\)

. This expression has several components: a coefficient (like 4 or 9), variables

• \(t\) and \(s\)

, and these variables are raised to powers.
Algebraic expressions can often be manipulated and simplified through methods such as factoring or expanding, using algebraic rules. This particular expression can be viewed as a combination of squares, making it an interesting candidate for special factoring techniques.

Polynomial Factorization

Polynomial factorization is the process of decomposing a polynomial into the product of other polynomials that, when multiplied together, give the original polynomial. This is a key skill in algebra because it allows you to simplify and solve polynomial equations more easily.
The given expression

• \(4t^2 - 9s^2\)

is a polynomial. Since it follows the difference of squares, we can apply the formula:

• \((a - b)(a + b) = a^2 - b^2\)

to factor it.
Here,

• \(a = 2t\)
• and \(b = 3s\)

. When you apply this formula, you get

• \((2t - 3s)(2t + 3s)\)

.
Successfully factoring the polynomial verifies your understanding and allows for further operations to be performed, such as solving equations or simply rewriting the expression in a new form.