Problem 6 - Elementary and Intermediate Algebra

Question

Fill in the blanks. a. \(36 x^{2}=(\quad)^{2}\) b. \(100 x^{4}=(\quad)^{2}\) c. \(4 x^{2}-9=(\quad)^{2}-(\quad)^{2}\)

Step-by-Step Solution

Answer

a) \((6x)\), b) \((10x^2)\), c) \((2x), (3)\).

Step 1: Factorize the Square in Expression 'a'

The given expression is \(36x^2=(\quad)^2\). Recognize that \(36x^2\) can be expressed as a perfect square. Note \(36 = 6^2\) and thus \(36x^2 = (6x)^2\). Therefore, the blank can be filled with \(6x\).

Step 2: Factorize the Square in Expression 'b'

The given expression is \(100x^4=(\quad)^2\). Recognize that \(100 = 10^2\) and \(x^4 = (x^2)^2\). Therefore, \(100x^4 = (10x^2)^2\). Hence, the blank is filled with \(10x^2\).

Step 3: Express Expression 'c' as a Difference of Squares

The given expression is \(4x^2 - 9 = (\quad)^2 - (\quad)^2\). Recognize that \(4x^2 = (2x)^2\) and \(9 = 3^2\). Thus, \(4x^2 - 9 = (2x)^2 - (3)^2\). Therefore, the blanks are filled with \(2x\) and \(3\), respectively.

Key Concepts

Difference of Squares

Understanding the concept of the difference of squares is crucial in algebra, particularly when dealing with polynomial expressions. The difference of squares is a specific algebraic identity and is written as
\[(a^2 - b^2) = (a + b)(a - b)\].
This means that whenever you have a subtraction of two squared terms, such as
\[x^2 - 9 = (x)^2 - (3)^2\], you can factor it into two separate binomials.

• Identify the two perfect squares, which in this case are \((x)^2\) and \((3)^2\).
• Apply the formula: Create two binomial expressions, \((x + 3)\) and \((x - 3)\).

The convenience of using the difference of squares formula is that it simplifies expressions quickly and makes equations easier to solve.
In exercises like "\(4x^2 - 9 = (2x)^2 - (3)^2\)," it helps to quickly identify that the expression can be split into \((2x + 3)(2x - 3)\).

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations. They act as building blocks in mathematics. Understanding them is essential for mastering algebra.
These expressions can include terms that vary greatly:

• Constants, like 3 or 9.
• Variables, such as \(x\) or \(y\).
• Coefficients, numbers multiplying variables, for example, 4 in 4x.

When crafting an algebraic expression, recognize the operations involved—addition, subtraction, multiplication, or division. An expression like \(4x^2 - 9\) combines multiple concepts:

• \(4x^2\) contains multiplication of a variable raised to a power.
• \(-9\) subtracts a constant value.

Recognizing and manipulating different parts of these expressions allows you to perform operations, factor, and simplify efficiently. This forms the basis of solving complex equations.

Polynomial Factorization

Polynomial factorization is a technique used to simplify expressions and solve polynomial equations. It involves breaking down a polynomial into a product of simpler polynomials or numbers.
Steps to factor a polynomial often include:

• Identifying common factors in the polynomial terms.
• Applying special factorization formulas, like the difference of squares.
• Checking for patterns or trinomials that can be factored into binomials.

Factorization turns complex expressions, like \(100x^4\), into simpler forms. For example, recognizing that \(100x^4 = (10x^2)^2\) shows that the polynomial is a perfect square.
This simplification:
- Makes it easier to solve equations and inequalities, - Offers insights into the roots or solutions of the polynomial. Mastering polynomial factorization is essential for tackling a wide range of algebraic challenges, from basic equation solving to advanced calculus.