Problem 209 - Elementary Algebra

Question

The height of a toy rocket launched with an initial speed of 80 feet per second from the balcony of an apartment building is related to the number of seconds, \(t,\) since it is launched by the trinomial \(-16 t^{2}+80 t+96\). Completely factor the trinomial.

Step-by-Step Solution

Answer

The factored trinomial is y = -16(t - 6)(t + 1).

Step 1: Identify and Write Down the Trinomial

Start with the given equation for the height of the toy rocket: y = -16t^2 + 80t + 96

Step 2: Identify the Greatest Common Factor (GCF)

Find the greatest common factor of the coefficients y = -16(t^2 - 5t - 6).

Step 3: Factor the Trinomial Inside the Parentheses

We need to factor the expression inside the parentheses: t^2 - 5t - 6. Look for two numbers that multiply to -6 and add to -5. These numbers are -6 and 1.

Step 4: Write the Factored Form

Using the factors identified in the previous step, rewrite the trinomial:t^2 - 5t - 6 = (t - 6)(t + 1).

Step 5: Combine with the GCF

Include the GCF identified in the second step to get the final factored form:y = -16 (t - 6)(t + 1).

Key Concepts

Greatest Common Factor (GCF)

To completely factor a trinomial, finding the Greatest Common Factor (GCF) is an important step.
The GCF is the largest factor that divides all terms in the expression.
In the given problem, we start with the trinomial: \(y = -16t^2 + 80t + 96\).
Here are the steps to find the GCF:

• List the factors of each term:
  -16: -1, -2, -4, -8, -16
  80: 1, 2, 4, 5, 8, 10, 16, 20, 40, 80
  96: 1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, 96
• Identify the greatest common factor among these:
  The GCF of -16, 80, and 96 is 16.

Using the GCF, we can factor out -16: \[y = -16(t^2 - 5t - 6)\].
This simplification makes it easier to factor the remaining trinomial.

Quadratic Equations

A quadratic equation is a polynomial equation of the form \(ax^2 + bx + c = 0\).
In this exercise, we're given the quadratic \(y = -16t^2 + 80t + 96\).
Understanding quadratic equations helps to identify key features:

• The highest degree (squared term) means it's a quadratic.
• The coefficients determine the parabola's shape.

Standard form, \(ax^2 + bx + c\), helps in solving by factoring or using formulas.
Here, after factoring out the GCF, we have \(t^2 - 5t - 6\).
The next step is to rewrite this trinomial in factored form.

Factored Form

The final key concept is writing the quadratic equation in factored form.
Factored form of a quadratic equation is \( (t - r)(t - s)\) where 'r' and 's' are the roots.
For the trinomial \(t^2 - 5t - 6\), find two numbers that multiply to -6 and add to -5.
These numbers are -6 and 1.
Therefore, \(t^2 - 5t - 6 = (t - 6)(t + 1)\).
Including the GCF, the fully factored form is: \[ y = -16(t - 6)(t + 1)\].
This process simplifies solving and understanding quadratic equations.
Factoring transforms a complex expression into usable roots.