Can Factoring Polynomials Help Us with Projectile Motion?

Yes, it can! Factoring polynomials is a key skill in Grade 10 Algebra I. It helps us not only make math easier but also solve real-world problems like those involving projectile motion. Let’s take a closer look.

What is Projectile Motion?

Projectile motion is the path of an object that’s thrown or shot into the air. This path is affected by things like gravity. We can describe the height of the object over time using math, specifically with quadratic equations.

A common way to write this equation looks like this:

$$h(t) = -at^2 + bt + c$$

In this equation:

• ( h(t) ) is the height of the object at time ( t ).
• ( a ), ( b ), and ( c ) are numbers we use in the equation.

The negative sign in front of ( at^2 ) tells us that the object will fall back down because of gravity.

How Does Factoring Help?

Factoring helps us figure out when the object hits the ground, which happens when the height ( h(t) ) is zero. So we set our equation to zero:

$$0 = -at^2 + bt + c$$

To solve this, we can factor the equation into simpler parts. Let’s look at an example:

$$h(t) = -5t^2 + 20t + 15$$

To find when ( h(t) = 0 ), we can factor this equation:

1. Identify the Numbers: For our example, we have ( a = -5 ), ( b = 20 ), and ( c = 15 ).

2. Factor Out Common Parts: We notice we can factor out -5:

   $$h(t) = -5(t^2 - 4t - 3)$$

3. Use Factoring Techniques: Now, we need to factor ( t^2 - 4t - 3 ):

   $$t^2 - 4t - 3 = (t - 5)(t + 1)$$

Now our equation looks like this:

$$0 = -5(t - 5)(t + 1)$$

Finding the Solutions

Next, we find the values for ( t ) when the height is zero:

$$-5(t - 5)(t + 1) = 0$$

This gives us two answers:

1. ( t - 5 = 0 ) → ( t = 5 )
2. ( t + 1 = 0 ) → ( t = -1 ) (we can't use this one because time can't be negative)

So, the object hits the ground at ( t = 5 ) seconds.

Why Is This Important?

Factoring polynomials helps us find out when a projectile will land. We can also find out things like the highest point it reaches by looking at the top point of the curve made by our quadratic equation. By factoring, we can simplify lots of calculations about how the object moves.

In Conclusion

Overall, factoring polynomials is super useful for understanding projectile motion. It helps us break down complex problems into easier parts. Whether it's figuring out when something hits the ground or how high it goes, knowing how to factor polynomials is key to using these ideas in science and engineering. So remember, factoring isn’t just another math trick; it’s a handy tool for solving real-world problems!