Problem 97 - College Algebra

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 described by the polynomial \(256-16 t^{2}\) Factor this expression completely.

Step-by-Step Solution

Answer

\(256 - 16t^2\) completely factors to \(16(4 - t)(4 + t)\)

Step 1: Identify the Greatest Common Factor

The first step in factoring an expression is to identify the greatest common factor (GCF). The GCF for our expression, \(256 - 16t^2\), is 16. The GCF can be factored out of the expression.

Step 2: Factor out the GCF

Factoring out the GCF of 16 gives us the reduced expression \(16(16 - t^2)\). We can go further by factoring the quadratic inside the parentheses.

Step 3: Factor the Quadratic

The quadratic \(16 - t^2\) can be factored as \((4 - t)(4 + t)\). This makes use of the difference of squares approach. The rule is \(a^2−b^2 = (a−b)(a+b)\). If we let \(a = 4\) and \(b = t\), it fits the pattern perfectly.

Step 4: State the Completely Factored Expression

Multiplying the GCF factored out in Step 2 with the quadratics factored in Step 3 gives the completely factored expression. As a result, the factored form of the expression is \(16(4 - t)(4 + t)\).

Step 5: Verify the Result

Expand the factored form of the expression to confirm that you obtain the original quadratic equation \(256 - 16t^2\). This step is done to check the accuracy of your factoring.

Key Concepts

Greatest Common Factor

Factoring polynomials often begins with identifying and factoring the greatest common factor (GCF). The GCF is the largest number that can divide each term in the polynomial without leaving a remainder. For instance, in the expression \(256 - 16t^2\), the greatest common factor is 16 because both 256 and 16 can be divided by 16.To factor the GCF out of the polynomial:

• First, divide each term by the GCF (16 in this case).
• This simplifies \(256 - 16t^2\) to \(16(16 - t^2)\).

Once the GCF is factored out, we are left with a simpler expression inside the parentheses, \(16 - t^2\), which can be further factored. This step is fundamental because it simplifies polynomials and sets the stage for additional factoring techniques.

Difference of Squares

The difference of squares is a special factoring technique used to simplify expressions where two square terms are subtracted. In the expression \(16 - t^2\), you can see that it's structured as a difference of squares. Specifically, \(16\) is \(4^2\) and \(t^2\) is \(t^2\), so it follows the pattern \(a^2 - b^2\). The general rule for the difference of squares is:\[a^2 - b^2 = (a-b)(a+b)\]Using this rule:

• Identify \(a = 4\) and \(b = t\).
• Factor \(16 - t^2\) as \((4 - t)(4 + t)\).

Recognizing the difference of squares allows for quick and straightforward factoring of the quadratic part of the expression, which ultimately helps in achieving the completely factored form.

Quadratic Expressions

Quadratic expressions are polynomial expressions of the form \(ax^2 + bx + c\), but in our context, the expression is simplified as \(16 - t^2\), fitting into a type of quadratic expression. Although it’s not a typical quadratic with a middle term, understanding its structure, particularly as a difference of squares, is vital for factoring.To factor a quadratic expression effectively:

• Look for patterns like the difference of squares, as seen in \(16 - t^2\).
• Recognize that these quadratics might be split into two binomials, as done in the case of \(16 - t^2\) becoming \((4-t)(4+t)\).

Quadratic expressions sometimes hide in simpler forms, making familiarity with factoring methods essential for solving polynomial problems. By mastering these techniques, factored forms like \(16(4 - t)(4 + t)\) become achievable through systematic steps.