Problem 52 - Elementary and Intermediate Algebra

Question

Factor. $$ 400 z^{2}-1 $$

Step-by-Step Solution

Answer

\((20z - 1)(20z + 1)\)

Step 1: Identify the Form

Recognize that the expression \(400z^{2} - 1\) is a difference of squares. The general form for the difference of squares is \(a^{2} - b^{2} = (a-b)(a+b)\).

Step 2: Express as Squares

Write each term as a squared number: \(400z^2 = (20z)^2\) and \(1 = 1^2\). Hence, the expression can be seen as \((20z)^2 - 1^2\).

Step 3: Apply the Difference of Squares Formula

Apply the formula \(a^{2} - b^{2} = (a-b)(a+b)\) to \((20z)^{2} - 1^{2}\), using \(a = 20z\) and \(b = 1\). Thus, \((20z - 1)(20z + 1)\).

Key Concepts

Difference of Squares

The concept of the Difference of Squares is a fundamental aspect of polynomial factoring. It refers to expressions that can be written in the form \(a^2 - b^2\), where both \(a^2\) and \(b^2\) are perfect squares. These expressions can be factored into \((a - b)(a + b)\). The difference of squares is called so because there is a subtraction (difference) between the two squared terms.Understanding difference of squares:- **Perfect Squares:** These are numbers or expressions that can be expressed as the product of an integer or expression with itself, such as \(4\), \(9z^2\), and \(16y^4\), which respectively are \(2^2\), \((3z)^2\), and \((4y^2)^2\).- **Identifying the Difference:** Look for the subtraction of two squared terms, like in the expression \(400z^2 - 1\), which can be rewritten as \((20z)^2 - 1^2\).This factoring technique is particularly useful because it simplifies expressions and aids in solving equations. Recognizing the difference of squares allows students to transform complex binomials into products of simpler binomials, thus making further algebraic manipulation more manageable.

Quadratic Expressions

Quadratic Expressions are polynomials that follow the general form \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants, and \(x\) is the variable. These expressions are characterized by their highest power being 2.While the original task didn't involve a full quadratic expression, the principles of dealing with quadratic parts of expressions are still relevant.Key Features of Quadratic Expressions:

• Standard Form: Expressed as \(ax^2 + bx + c\), it outlines a parabolic curve when graphed.
• Factoring: Finding factors of the quadratic expression can often simplify solving for \(x\).
• Vertex and Roots: They can be crucial in graphing and understanding the expression's behavior.

For instance, in the difference of squares \(400z^2 - 1\), each term can be seen as a simplified quadratic where full factorization principles help rearrange the terms like addressing the squared \(z\) factor. Having command over quadratic expressions lends itself well when engaged in higher-level polynomial factoring.

Polynomial Factoring Techniques

Polynomial Factoring Techniques encompass a range of strategies used to break down polynomials into products of simpler polynomials. These techniques are critical for simplifying expressions, solving polynomial equations, and performing algebraic operations.Key techniques in polynomial factoring include:- **GCF (Greatest Common Factor):** Always check if there's a common factor in all terms of the polynomial.- **Difference of Squares:** As illustrated earlier, you can factor polynomials of the form \(a^2 - b^2\) into \((a-b)(a+b)\).- **Factoring Quadratics:** Quadratic polynomials can often be factored into products of binomials by identifying numbers that multiply to \(ac\) and add to \(b\).- **Synthetic Division and Polynomial Long Division:** These methods help factor larger polynomials by dividing and revealing factors.For the specific exercise of \(400z^2 - 1\), recognizing the difference of squares allowed us to directly apply a straightforward technique, showcasing the power and simplicity of these methods. Understanding these principles not only aids in correctly solving algebra problems but also provides a toolkit for tackling a variety of mathematical challenges involving polynomials.