Robot Exploration: Solve for Return Times Using Cubic Equations

What This Problem Teaches

• How to solve cubic equations by factoring out common terms
• When a polynomial equals zero, each factor must equal zero
• How to identify and solve quadratic equations within cubic problems
• Physical interpretation of mathematical roots in motion problems
• Recognition that cubic polynomials can have up to three real solutions

Visualizing the Robot's Journey

This graph shows the robot's distance from home base over time. The red dots mark when the robot is at home base (distance = 0). Notice how the robot ventures out, returns briefly at t = 5 hours, then ventures out again before returning finally at t = 8 hours.

Solution: Method 1 — Factor and Conquer

The robot returns to its home base when its distance from base equals zero. We need to solve d(t) = 0.

Step 1 — Set up the equation

We need to find all values of t where:

Step 2 — Factor out the common term

Notice that every term contains t as a factor. Factor it out:

This immediately gives us one solution: t = 0 (the robot starts at home base).

Step 3 — Solve the quadratic factor

Now we need to solve t² − 13t + 40 = 0. Let's look for factors of 40 that add to 13:

• 40 = 1 × 40, and 1 + 40 = 41 (too big)
• 40 = 2 × 20, and 2 + 20 = 22 (too big)
• 40 = 4 × 10, and 4 + 10 = 14 (close!)
• 40 = 5 × 8, and 5 + 8 = 13 ✓

So we can factor: t² − 13t + 40 = (t − 5)(t − 8)

Step 4 — Complete the factorization

Step 5 — Find all solutions

For a product to equal zero, at least one factor must be zero:

• t = 0 (robot starts at home)
• t − 5 = 0, so t = 5
• t − 8 = 0, so t = 8

Solution: Method 2 — Using the Quadratic Formula

After factoring out t, we can also solve t² − 13t + 40 = 0 using the quadratic formula as verification.

Step 1 — Apply the quadratic formula

For t² − 13t + 40 = 0, we have a = 1, b = −13, c = 40:

Step 2 — Calculate both solutions

Step 3 — Combine with the factored solution

Including the t = 0 solution from factoring out t, our complete solution set is t = 0, 5, 8.

Verification

Let's verify each solution by substituting back into the original equation:

Check t = 0:

Check t = 5:

Check t = 8:

All three solutions check out perfectly!

Watch Out For These

The Pattern Behind This

This problem illustrates a key strategy for cubic equations: always check for common factors first. When every term contains the variable, you can factor it out immediately, reducing a cubic to a quadratic.

This gives you one solution (t = 0) instantly and reduces the problem to solving a quadratic equation. The general pattern is:

1. Factor out the greatest common factor
2. Solve each factor equal to zero
3. Collect all solutions

For motion problems specifically, t = 0 often represents the starting position or initial time, while the other solutions represent specific moments when certain conditions are met.

Real Applications

• Projectile Motion: The height of a thrown ball follows a cubic or quadratic path. Finding when height equals ground level gives you launch and landing times.
• Economics: Profit functions often involve cubic polynomials. Setting profit to zero finds break-even points for a business.
• Engineering: Structural deflection under varying loads can be modeled by cubic equations. Zero deflection points indicate optimal support locations.