Polynomial and Rational Functions

Polynomial End Behavior

Determine the end behavior of p(x) = 2x³ - x + 5 from its degree and leading coefficient without computing specific values.

Odd degree, positive leading coefficient: As x → +∞: p → +∞. As x → -∞: p → -∞. Odd degree, negative leading coefficient: As x → +∞: p → -∞. As x → -∞: p → +∞. Even degree, positive leading coefficient: both ends go to +∞. Even degree, negative leading coefficient: both ends go to -∞. Example: p(x) = 2x³ - x + 5. Degree 3 (odd), leading coefficient 2 (positive). So p → +∞ as x → +∞ and p → -∞ as x → -∞.

Example

Use degree parity and leading coefficient sign to determine end behavior.

highlighted = computed this step

Step 1 — Set up

Set up the expression.

p(x)=2x3−x+5p(x)= 2x^{3} -x+ 5

Step 2 — Leading term

The leading term is 2x to the 3.

2x3\hlmath{2x^{3}}

Step 3 — Degree and sign

Degree 3 is odd and leading coefficient 2 is positive.

degree 3,positive leading coefficient 2\text{degree } \hl{3} ,\quad\text{positive leading coefficient } \hl{2}

Step 4 — Right end

Odd degree with positive lead rises to the right.

x→+∞,p(x)→positive infinityx\to+\infty,\quad p(x)\to \hl{positive infinity}

Step 5 — Left end

Odd degree with positive lead falls to the left.

x→−∞,p(x)→negative infinityx\to-\infty,\quad p(x)\to \hl{negative infinity}
polynomial-end-behavior End behavior describes how a polynomial behaves as x → ±∞. It depends only on the degree and leading coefficient: