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Leading Coefficient Test

The Leading Coefficient Test is a method for determining the end behavior of a polynomial function by looking at two things: whether the degree is even or odd, and whether the leading coefficient is positive or negative.

The Leading Coefficient Test states that for a polynomial function f(x)=anxn+an−1xn−1+⋯+a1x+a0f(x) = a_nx^n + a_{n-1}x^{n-1} + \cdots + a_1x + a_0, the end behavior as x→±∞x \to \pm\infty depends solely on the leading term anxna_nx^n. If nn is even and an>0a_n > 0, both ends rise. If nn is even and an<0a_n < 0, both ends fall. If nn is odd and an>0a_n > 0, the left end falls and the right end rises. If nn is odd and an<0a_n < 0, the left end rises and the right end falls.

Worked Example

Problem: Use the Leading Coefficient Test to describe the end behavior of f(x)=−3x4+5x3−2x+7f(x) = -3x^4 + 5x^3 - 2x + 7.
Step 1: Identify the leading term. The term with the highest power of xx is the leading term.
Leading term=−3x4\text{Leading term} = -3x^4
Step 2: Determine the degree. The exponent on the leading term is 4, which is even.
n=4(even)n = 4 \quad (\text{even})
Step 3: Determine the leading coefficient and its sign. The coefficient of x4x^4 is −3-3, which is negative.
an=−3(negative)a_n = -3 \quad (\text{negative})
Step 4: Apply the test: even degree with a negative leading coefficient means both ends of the graph point downward.
As x→−∞,  f(x)→−∞andas x→+∞,  f(x)→−∞\text{As } x \to -\infty,\; f(x) \to -\infty \quad \text{and} \quad \text{as } x \to +\infty,\; f(x) \to -\infty
Answer: Both ends of the graph fall. As xx approaches −∞-\infty or +∞+\infty, f(x)f(x) approaches −∞-\infty.

Why It Matters

The Leading Coefficient Test gives you a quick way to sketch the overall shape of any polynomial without plotting many points. Engineers and scientists use end behavior to understand how models behave at extreme values — for instance, predicting whether a profit function eventually increases or decreases as production scales up. It also helps you verify that a graph you've drawn or a calculator has produced makes sense for the given equation.

Common Mistakes

Mistake: Using a term other than the one with the highest exponent as the leading term, especially when the polynomial isn't written in standard form.
Correction: Always identify the term with the largest exponent regardless of where it appears in the expression. For example, in 2x+7x3−x22x + 7x^3 - x^2, the leading term is 7x37x^3, not 2x2x.
Mistake: Confusing the rules for odd and even degree — thinking that odd degree means both ends go the same direction.
Correction: Odd-degree polynomials always have ends that go in opposite directions (one up, one down). Even-degree polynomials have ends that go in the same direction (both up or both down).

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