Problem 31 - College Algebra with Corequisite Support

Question

For the following exercises, multiply the binomials. \((4 c+1)(4 c-1)\)

Step-by-Step Solution

Answer

The product is \(16c^2 - 1\).

Step 1: Identify the Formula

The given expression \((4c + 1)(4c - 1)\) is recognized as a difference of squares. It follows the formula \((a + b)(a - b) = a^2 - b^2\).

Step 2: Apply the Difference of Squares Formula

Using the formula \((a + b)(a - b) = a^2 - b^2\), identify \(a = 4c\) and \(b = 1\). Substitute these values into the formula to get \((4c)^2 - 1^2\).

Step 3: Simplify the Expressions

Calculate \((4c)^2 = 16c^2\) and \(1^2 = 1\). After substituting these values, the expression simplifies to \(16c^2 - 1\).

Key Concepts

Difference of Squares

One important algebraic identity is the Difference of Squares. This identity is used in expressions where two binomials have the form

• One binomial in the form of \((a + b)\).
• Another binomial in the form of \((a - b)\).

The Difference of Squares formula is expressed as:\[(a + b)(a - b) = a^2 - b^2\]Here,

• \(a^2\) means \(a\) squared,
• And \(b^2\) means \(b\) squared.

The concept is that when you multiply these specific forms of binomials, the middle terms cancel each other out, and you're left with the square of each term being subtracted. This powerful tool greatly simplifies the multiplication of specific polynomial expressions like \((4c + 1)(4c - 1)\), allowing us to efficiently find the result \(16c^2 - 1\). Remembering this identity can save time and effort when faced with similarly structured problems.

Polynomial Expressions

Polynomial expressions consist of variables raised to whole number powers and coefficients.

• Each term in a polynomial is a product of a constant, called a coefficient, and a variable raised to an exponent.
• For example, in the polynomial term \(16c^2\), 16 is the coefficient, and \(c^2\) is the variable portion.

In general, a polynomial may have one or multiple terms, and mathematically, it can be represented as:

• A single term polynomial, such as \(ax^n\)
• A multi-term polynomial, such as \(ax^n + bx^{n-1} + \,\ldots\,+ z\)

When multiplying polynomials, it is often required to apply distributive properties or recognized algebraic identities such as the 'Difference of Squares' to simplify expressions. Always ensure to combine like terms and simplify the expression as shown in \(16c^2 - 1\). This results from multiplying the binomials \((4c + 1)(4c - 1)\), utilizing the polynomial and algebra identities effectively.

Algebraic Identities

In algebra, certain standard equations, termed algebraic identities, vastly simplify calculation. They provide set paths to come to conclusions without needing to rely solely on step-by-step evaluation. Among such fundamental identities is the Difference of Squares, which allows quick simplification of particular forms of polynomial expressions.

Many algebraic identities involve the use of binomials and further can be extended to work with trinomials and polynomials of higher degree. Learning these formulas is akin to understanding rules that govern arithmetic but in a more abstract form.

• By recognizing patterns, like the Difference of Squares or Sum and Difference of Cubes, you can swiftly evaluate and expand polynomials.
• Using algebraic identities can sometimes lead to discovering unexpected results or shortcuts, proving highly beneficial in problem-solving scenarios.

For example, in situations like solving \((4c + 1)(4c - 1) = 16c^2 - 1\), utilizing identities cuts through complexity by focusing on the core symmetrical nature of the expression, providing clear and efficient solutions.