Problem 28 - Elementary and Intermediate Algebra

Question

Factor difference of two squares. \(4 a^{2} b^{4}-9 d^{6}\)

Step-by-Step Solution

Answer

The factored form is \((2ab^2 + 3d^3)(2ab^2 - 3d^3)\).

Step 1: Identify the Form

Recognize that the given expression is a difference of two squares, which takes the form of \(A^2 - B^2\). Our task is to rewrite the expression so it matches this form.

Step 2: Express Each Term as a Square

Observe the terms in the expression \(4 a^{2} b^{4}-9 d^{6}\). Rewrite each term as a square: \((2ab^2)^2 - (3d^3)^2\).

Step 3: Apply the Difference of Squares Formula

Use the formula \(A^2 - B^2 = (A+B)(A-B)\). In our expression, consider \(A = 2ab^2\) and \(B = 3d^3\). Apply the formula: \((2ab^2 + 3d^3)(2ab^2 - 3d^3)\).

Step 4: Verify the Factorization

Expand \((2ab^2 + 3d^3)(2ab^2 - 3d^3)\) to check the correctness. The expansion gives \((2ab^2)^2 - (3d^3)^2\), which matches the original expression.

Key Concepts

Factoring Polynomials

Factoring polynomials is a fundamental concept in algebra. It involves breaking down a complex polynomial into simpler, more manageable components or factors. In the context of the difference of squares, factoring becomes quite straightforward due to a specific formula.

• The expression must be in the form \(A^2 - B^2\).
• This is known as the difference of squares because two terms are subtracted, each of which is a perfect square.

To factor a polynomial, identify these squares and apply the difference of squares formula: \(A^2 - B^2 = (A + B)(A - B)\). For example, in the expression \(4a^2b^4 - 9d^6\), we decompose it to \((2ab^2)^2 - (3d^3)^2\). This sets the stage for using the formula, ultimately simplifying the expression into its factored form, \((2ab^2 + 3d^3)(2ab^2 - 3d^3)\). The ability to recognize and utilize these patterns is crucial in algebra.

Algebraic Expressions

Algebraic expressions form the backbone of algebra. They are combinations of variables, constants, and operations. Understanding how to manipulate these expressions is key to solving algebra problems.
In algebra, expressions like \(4a^2b^4 - 9d^6\) are common. Such expressions can often be rewritten to reveal underlying mathematical structures. Recognizing these forms allows you to apply strategies such as factoring.

• Variables represent unknown values and can be adjusted based on the problem's context.
• Constants are fixed numbers that do not change, providing stability to the expression.
• Operations include addition, subtraction, multiplication, and more, which are applied in expression manipulation.

The expression \(4a^2b^4 - 9d^6\) consists of variables \(a, b,\) and \(d\), multiplied and squared, demonstrating how power and multiplicative relationships drive the complexity of algebraic expressions.

Mathematical Identities

Mathematical identities are equations that hold true for all possible values of the involved variables. They are powerful tools in algebra that simplify expressions and solve equations efficiently.
The difference of squares is an essential identity in algebra. The formula \(A^2 - B^2 = (A + B)(A - B)\) is a prime example. It simplifies expressions with squared terms beautifully.

• Identities are universally reliable, no matter the values substituted into the equation.
• They provide shortcuts that enhance calculation speed and accuracy.
• Recognizing when an expression fits an identity can make solving algebra problems immensely easier.

In our exercise, the expression \((2ab^2)^2 - (3d^3)^2\) serves as an instance where recognizing the difference of squares identity simplifies the factorization process. Using identities like these is a vital skill for anyone learning algebra.