Problem 312 - Elementary Algebra

Question

A fall tradition at the University of California San Diego is the Pumpkin Drop, where a pumpkin is dropped from the eleventh story of Tioga Hall. (a) The binomial \(-16 t^{2}+128\) gives the height of the pumpkin \(t\) seconds after it is dropped. Factor the greatest common factor from this binomial. (b) If the pumpkin is thrown down with initial velocity 32 feet per second, its height after \(t\) seconds is given by the trinomial \(-16 t^{2}-32 t+128\). Completely factor this trinomial.

Step-by-Step Solution

Answer

(a) \[ -16(t^2 - 8) \], (b) \[ -16(t + 4)(t - 2) \].

Step 1: Identify the Greatest Common Factor (GCF)

Look at the binomial \( -16t^2 + 128 \). Identify the common factor that can be factored out of both terms.

Step 2: Factor out the GCF

The GCF of \( -16t^2 \) and \( 128 \) is 16. Factor out 16 from \( -16t^2 + 128 \): \[ -16t^2 + 128 = -16(t^2 - 8) \].

Step 3: Determine the Trinomial

Write down the trinomial given in the problem: \[ -16t^2 - 32t + 128 \].

Step 4: Identify the GCF of the Trinomial

Determine the GCF for all three terms in the trinomial. The GCF is 16.

Step 5: Factor out the GCF from the Trinomial

Factor 16 out from the trinomial: \[ -16t^2 - 32t + 128 = -16(t^2 + 2t - 8) \].

Step 6: Factor the Remaining Trinomial

To factor \( t^2 + 2t - 8 \), find two numbers that multiply to -8 and add to 2. These numbers are 4 and -2. So, \[ t^2 + 2t - 8 = (t + 4)(t - 2) \].

Step 7: Combine All Factors

Combine the factored terms: \[ -16(t^2 + 2t - 8) = -16(t + 4)(t - 2) \].

Key Concepts

Understanding the Greatest Common Factor (GCF)

The Greatest Common Factor, or GCF, is the largest number that can evenly divide both terms in a polynomial. Finding the GCF is crucial because it simplifies expressions, making them easier to work with.

To find it, follow these steps:

• List the factors of each term in the polynomial.
• Identify the largest common factor shared by each term.

Let's take the binomial \(-16t^2 + 128\). The factors of \(-16t^2\) are 1, 2, 4, 8, and 16, along with their negative counterparts. The factors of 128 are likewise 1, 2, 4, 8, and 16. Thus, the GCF of both terms is 16. We can factor out this GCF:
\[ -16t^2 + 128 = -16(t^2 - 8) \]

By factoring out the GCF, we've simplified the original expression, making it easier to solve or manipulate.

Factoring Trinomials

Factoring trinomials involves breaking down an algebraic expression into products of simpler expressions. Trinomials take the form \(ax^2 + bx + c\). Finding factors means choosing two binomials \((d + e)(f + g)\) whose product equals the original trinomial.

Here’s how you can tackle the trinomial \(-16t^2 - 32t + 128\):

• First, identify the GCF, which is 16 for all three terms.
• Factor out the GCF:
  \[ -16t^2 - 32t + 128 = -16(t^2 + 2t - 8) \]

Next, you need to find two numbers that multiply to -8 and add up to 2 (the coefficient of the middle term). These numbers are 4 and -2. Now, split the middle term based on these factors and factor by grouping:
\[ t^2 + 2t - 8 = (t+4)(t-2) \]

Combine all parts to get the final factored form:
\[ -16(t^2 + 2t - 8) = -16(t + 4)(t - 2) \]

Breaking Down Binomial Expressions

Binomial expressions consist of two terms separated by a plus or minus sign. Understanding binomials is a stepping stone to mastering more complex algebraic expressions.

Consider the binomial given in part (a) of the exercise: \(-16t^2 + 128\). Recalling that the GCF is 16, we factor this out:
\[ -16t^2 + 128 = -16(t^2 - 8) \]
Here, we've simplified the binomial, making it easier to interpret and use in further calculations.

Factoring out the GCF essentially 'shrinks' the equation. It zeros in on the crucial parts needed to solve or simplify the problem.
Understanding binomials prepares you for more comprehensive concepts like trinomials and polynomial factorizations.