Investigate whether a cubic function and a linear function can exist without any points of intersection | Step-by-Step Solution

What We're Solving:

We need to figure out whether it's possible for a cubic function (degree 3 polynomial) and a linear function (degree 1 polynomial) to never cross each other - that is, have zero points where they intersect.

The Approach:

To solve this, we'll think about what "intersection" means mathematically, then use some key properties of polynomials. When two functions intersect, they have the same y-value at that x-coordinate. So we're really asking: can the equation "cubic = linear" have no solutions?

Step-by-Step Solution:

Step 1: Set up the intersection equation If we have a cubic function f(x) = ax³ + bx² + cx + d and a linear function g(x) = mx + n, they intersect where: ax³ + bx² + cx + d = mx + n

Step 2: Rearrange to standard form Moving everything to one side: ax³ + bx² + cx + d - mx - n = 0 ax³ + bx² + (c-m)x + (d-n) = 0

Step 3: Recognize what we have This is still a cubic equation (since a ≠ 0 for a true cubic function). Now here's the key insight: we need to determine if a cubic equation can have zero real solutions.

Step 4: Use the Fundamental Theorem of Algebra A cubic polynomial has exactly 3 roots (counting multiplicities), but some might be complex numbers. The question is: can ALL three roots be complex (non-real)?

Step 5: Apply the Complex Conjugate Root Theorem Here's the crucial point! For polynomials with real coefficients, complex roots must come in conjugate pairs. Since 3 is an odd number, we cannot pair up all three roots. This means at least one root must be real.

Step 6: Draw the conclusion Since our intersection equation ax³ + bx² + (c-m)x + (d-n) = 0 must have at least one real solution, the cubic and linear functions must intersect at least once.

The Answer:

No, a cubic function and a linear function cannot have zero intersections. They must intersect at least once, and could intersect up to three times.

Memory Tip:

Remember "Odd degrees can't hide!" - Polynomial equations with odd degrees (like cubic equations) always have at least one real root, so a cubic and linear function must meet somewhere on the real coordinate plane.

You're doing great by thinking about these fundamental relationships between functions! This type of reasoning - connecting intersection points to polynomial roots - is a powerful tool you'll use throughout algebra and calculus.