Problem 35 - College Algebra

Question

In Exercises \(31-40,\) factor the difference of two squares. $$9 x^{2}-25 y^{2}$$

Step-by-Step Solution

Answer

The factored form of the expression \(9x^{2} - 25y^{2}\) is \((3x - 5y)(3x + 5y)\).

Step 1: Identify a and b

In the expression \(9x^{2} - 25y^{2}\), identify a and b from the formula \(a^{2} - b^{2}\). In this case, \(a^{2}\) is \(9x^{2}\) which gives us that \(a = 3x\), and \(b^{2}\) is \(25y^{2}\), which gives us that \(b = 5y\).

Step 2: Use the difference of two squares formula

Substitute a and b into the formula \((a - b)(a + b)\) to factorize the expression. This gives us \((3x - 5y)(3x + 5y)\).

Step 3: Write the final solution

The factored form of the given expression \(9x^{2} - 25y^{2}\) is \((3x - 5y)(3x + 5y)\). Write down this as the final answer.

Key Concepts

Factoring Expressions

Factoring expressions is a fundamental skill in algebra that involves breaking down a composite mathematical expression into a product of simpler factors. This makes it easier to evaluate or simplify expressions and solve equations.

A common scenario in algebra is recognizing specific forms like the difference of squares. The expression \(9x^2 - 25y^2\) can be identified as a difference of two squares because both terms separately form perfect squares.

• \(9x^2\) is \((3x)^2\)
• \(25y^2\) is \((5y)^2\)

Once these squares are identified, they guide us into using the difference of squares formula to factor the expression efficiently.

Polynomial Identities

Polynomial identities provide a bridge to understanding how different polynomials behave under various operations, like addition, subtraction, and multiplication. One key identity that emerges often is the difference of squares identity.

When we write \(a^2 - b^2\), we recognize that it can directly be transformed using the identity:
\[(a - b)(a + b)\]
This is the basis for the factoring process observed in the expression \(9x^2 - 25y^2\).

• Identifying \(a = 3x\) and \(b = 5y\) allows us to utilize this identity directly.
• The expression simplifies and factors into two linear binomials, \((3x - 5y)(3x + 5y)\).

These identities simplify complex expressions and make them easier to work with, supporting both algebraic manipulation and problem-solving.

Algebraic Formulas

Algebraic formulas like the difference of squares are key tools in simplifying and solving algebraic expressions and equations. These formulas are precise instructions for transforming one expression into another in a way that is mathematically valid and useful.

The specific formula for the difference of squares is:\[a^2 - b^2 = (a - b)(a + b)\]This formula works because it represents the geometric difference of the two square areas, allowing intercepts that simplify expressions easily.

• This helps in breaking down cubic or quartic equations into more manageable forms.
• It plays a crucial role not only for factoring but also in roots finding.

Understanding and applying algebraic formulas effectively equips students with the skills to tackle a range of mathematical problems, simplifying tasks and reducing potential arithmetic errors.