Question

Challenge Problems.$$9 a^{2} b^{2}-4 c^{4}$$

Short answer

\((3ab + 2c^2)(3ab - 2c^2)\)

Step-by-step solution

Identify the Pattern

Recognize that the expression is in the form of a difference of squares, which can be factored into \(a^2 - b^2 = (a-b)(a+b)\).

Apply the Difference of Squares Formula

Notice that \(9 a^{2} b^{2}\) is a perfect square \(\left(3ab\right)^2\) and \(4 c^{4}\) is also a perfect square \(\left(2c^2\right)^2\). Apply the difference of squares formula to write the expression as \(\left(3ab + 2c^2\right)\left(3ab - 2c^2\right)\).

Key concepts

Factoring Polynomials

Factoring polynomials is a critical skill in algebra, allowing students to break down complex algebraic expressions into simpler, more manageable pieces. Imagine it like dismantling a complex structure into its building blocks. The goal of factoring is to rewrite a polynomial as a product of its factors, much like expressing a number like 15 as the product of its factors, 3 and 5.

When it comes to factoring, it is essential to look for a Greatest Common Factor (GCF) first, which is the highest expression that divides all the terms in the polynomial. Once the GCF is factored out, other techniques like the difference of squares, sum and difference of cubes, or factoring trinomials come into play.

• For example, in the polynomial x^2 - 9, which is a difference of squares, it can be factored into (x + 3)(x - 3).
• On the other hand, a trinomial such as x^2 + 5x + 6 can often be factored into binomials like (x + 3)(x + 2), assuming the polynomial can be factored over the integers.

Focusing on recognizing patterns helps demystify the process, making it easier to decide which factoring method to employ.

Perfect Square Trinomials

A perfect square trinomial is formed by squaring a binomial. It always follows a particular pattern: a^2 + 2ab + b^2, which factors into (a + b)^2, or a^2 - 2ab + b^2, which factors into (a - b)^2. Understanding how to both recognize and factor these perfect squares is an important step in manipulating algebraic expressions.

The coefficients in a perfect square trinomial are particularly telling. The first and last terms are always perfect squares themselves, and the middle term is twice the product of the square roots of the first and last terms.

• For instance, the expression x^2 + 6x + 9 is a perfect square trinomial because it can be rewritten as (x + 3)^2.
• Similarly, 4x^2 - 12x + 9 is the perfect square (2x - 3)^2.

Algebraic Expressions

At the heart of algebra lies the use of algebraic expressions, which are combinations of numbers, variables, and arithmetic operations. These expressions can take many forms, from simple constants to complicated polynomials and functions. Understanding how to work with these expressions—and manipulate them through various laws and properties—is fundamental to solving equations and inequities.

Algebraic expressions can be simplified, expanded, factored, and evaluated. For instance, when given an expression like 7(2x + 3), you can distribute the 7 to both terms within the parentheses to get 14x + 21. This is called expanding the expression. Conversely, recognizing a common factor in multiple terms allows one to go in the opposite direction, condensing the expression into a more compact form.

These skills are especially useful when working with equations. By performing equivalent transformations—like adding the same amount to both sides or multiplying both sides by a non-zero number—we can isolate variables and solve for unknowns. Ultimately, gaining fluency in algebraic expressions is like learning a new language that describes mathematical relationships.