Problem 123 - Introductory Algebra for College Students

Question

A rock is dropped from the top of a 256 -foot cliff. The height, in feet, of the rock above the water after \(t\) seconds is modeled by the polynomial \(256-16 t^{2} .\) Factor this expression completely. (Image can't copy)

Step-by-Step Solution

Answer

The completely factored form of the polynomial \(256 - 16t^2\) is \(16(16t + 1)(16t - 1)\).

Step 1: Identify difference of squares

The polynomial \(256 - 16t^2\) can be written as \((16t)^2 - 16^2\), which is a difference of squares. In general, difference of squares \(a^2 - b^2\) can be factored as \((a+b)(a-b)\).

Step 2: Apply the difference of squares formula

Applying the difference of squares factoring formula \((a+b)(a-b)\) for our polynomial gives us \((16t + 16)(16t - 16)\).

Step 3: Simplify the factors

Each factor has a common factor of 16, which can be factored out. Factoring out the common factors results in \(16(16t + 1)16t - 1)\) as the factored form of the polynomial.

Key Concepts

Difference of Squares

The difference of squares is a mathematical concept used to factor expressions that take the form of one square subtracted from another, like \(a^2 - b^2\). When you encounter an algebraic expression that fits this pattern, the expression can be factored into the product of two binomials: \(a+b\) and \(a-b\).

This concept helps in simplifying expressions and solving equations. For instance, if we have the polynomial \(256 - 16t^2\), we recognize that \(256\) is \(16^2\) and \(16t^2\) is \(\left(16t\right)^2\), showcasing a perfect example of the difference of squares. Using the formula \(a^2 - b^2 = (a+b)(a-b)\), we factor it as \(\left(16t + 16\right)\left(16t - 16\right)\).

It is a useful technique in algebra, and can often be applied to polynomials in quadratic form to simplify them prior to solving or graphing.

Polynomial Factorization

Polynomial factorization is the process of deconstructing a polynomial, which is a sum of monomials, into a product of simpler polynomials that when multiplied together will give you the original polynomial. Consider it like finding the building blocks of the polynomial.

For example, the polynomial in our exercise is \(256 - 16t^2\). After determining that it is a difference of squares, we then want to factor it completely. This involves identifying common factors in the terms of the polynomial and then using algebraic identities, such as the difference of squares or other factoring techniques like grouping or long division.

Upon applying these methods, you get a more digestible expression, which in this case further simplifies by factoring out the greatest common factor of 16, resulting in \(16(16t + 1)(16t - 1)\). Polynomial factorization is crucial for simplifying expressions and solving polynomial equations.

Algebraic Expressions

An algebraic expression is a combination of numbers, variables, and arithmetic operations (like addition, subtraction, multiplication, division, and exponentiation by a variable amount). There are no equals signs in an algebraic expression, as that would make it an equation.

For instance, the expression \(256 - 16t^2\) in our problem represents the height of a rock above the water after \(t\) seconds, with \(256\) representing the initial height and \(16t^2\) representing the distance fallen due to gravity.

Understanding algebraic expressions is fundamental to algebra, as it forms the basis of equations and functions that model real-world problems. After interpreting what an expression represents, such as physical height over time in this case, we can manipulate and factor these expressions to explore their properties or solve for unknown quantities.