Problem 103 - College Algebra with Modeling and Visualization

Question

Factor the expression completely. \(16 x^{2}-25\)

Step-by-Step Solution

Answer

The expression factors to \((4x - 5)(4x + 5)\).

Step 1: Recognize the Difference of Squares

The expression given is in the form of the difference of squares: \(a^2 - b^2\), where \(a = 4x\) and \(b = 5\).

Step 2: Apply the Difference of Squares Formula

The formula for factoring a difference of squares is \(a^2 - b^2 = (a - b)(a + b)\). Substitute \(a = 4x\) and \(b = 5\) into the formula to get \((4x - 5)(4x + 5)\).

Step 3: Confirm the Factorization

Multiply the factors \((4x - 5)(4x + 5)\) to ensure they simplify back to the original expression:\[(4x - 5)(4x + 5) = (4x)^2 - (5)^2 = 16x^2 - 25\].This confirms our factorization is correct.

Key Concepts

Difference of Squares

The difference of squares is a key algebraic concept that involves recognizing and factoring expressions of the form \(a^2 - b^2\). In this context, \(a\) and \(b\) are any algebraic expressions or numbers, and they are each squared individually. The difference of squares formula simplifies \(a^2 - b^2\) into \((a - b)(a + b)\). This unique factorization arises because the middle terms cancel out when expanding the binomials.

For example, in the expression \(16x^2 - 25\), by identifying that \(16x^2\) is \((4x)^2\) and \(25\) is \(5^2\), we spot this as a difference of squares.

• The expression has a subtraction sign, indicating a 'difference'.
• Both terms are perfect squares: \(16x^2\) and \(25\).

Recognizing these qualities allows one to straightforwardly apply the formula and factor it efficiently. This fundamental technique is often used in simplifying polygons and solving quadratic equations.

Factorization

Factorization in algebra refers to the process of breaking down complicated expressions into simpler ones that, when multiplied together, produce the original expression. This makes calculations easier and simplifies solving equations. For example, any number or expression can be expressed as a product of its factors.

When you factorize a difference of squares, like \(16x^2 - 25\), you're identifying two binomials: \((4x - 5)\) and \((4x + 5)\). These binomials multiplied together give you the original quadratic expression. This method of expressing an expression as a product of its factors helps solve equations by setting each factor equal to zero and solving for the variable.

• This is particularly useful in quadratic equations where solutions can be found by factorizing the quadratic expression into binomials.
• This helps simplify complex algebraic expressions and makes math more manageable.

In summary, factorization is a valuable tool for simplifying problems and finding solutions.

Polynomials

Polynomials are expressions that consist of variables and coefficients, constructed using operations like addition, subtraction, and multiplication. They are fundamental structures in algebra, with terms consisting of variables raised to whole-number exponents. A typical polynomial, for example, looks like \(3x^3 + 2x^2 - 5x + 7\).

In the specific context of the original example, \(16x^2 - 25\) is a polynomial of degree 2, known as a quadratic polynomial. Quadratic polynomials like this one can often be simplified using techniques such as recognizing patterns like the difference of squares.

• The degree of a polynomial indicates the highest power of the variable; here, it's 2.
• Polynomials can include terms which are constants, like \(-25\), or multiples of variables raised to a power, like \(16x^2\).

Understanding and working with polynomials is crucial, providing foundation skills needed in further algebra and calculus courses. They not only offer a way to express numbers but also serve as tools for analyzing and solving algebraic equations.