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All courses Algebra II · A-SSE.1.a 21 of 55
Interpret terms, factors, and coefficients in polynomial and rational expressions.

Read leading coefficient, parity, and both polynomial ends

Problem
For \(p(x)~=~-3x^4~+~2x~-~1\), state the leading term, its sign and degree parity, and the behavior of \(\text{both}~\text{ends}\).
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Hint

Locate the term with the greatest exponent before reading any coefficient.

Leading coefficient is the coefficient of the highest-power term. Degree parity determines same/opposite ends; sign orients them.

Solution walkthrough

01

Identify the dominant term

\[p(x)=-3x^4+2x-1~->~\text{leading}~\text{term}=-3x^4\]

The term with the greatest exponent is negative 3x to the fourth. Lower-degree terms become negligible compared with it for large absolute x.

02

Read sign and parity

\[\text{leading}~\text{coefficient}=-3<0;~\text{degree}=4~\text{is}~\text{even}\]

Even degree means the two ends have the same direction; the negative leading coefficient makes that shared direction downward.

03

Determine the right end

\[x->+\text{infinity}~->~-3x^4->-\text{infinity}~->~p(x)->-\text{infinity}\]

A large positive x has a large positive fourth power, and multiplication by negative 3 sends the dominant term downward.

04

State both ends

\[\text{The}~\text{leading}~\text{term}~\text{is}~-3x^4,~\text{with}~\text{negative}~\text{even}~\text{degree}.~\text{Thus}~p(x)->-\text{infinity}~\text{as}~x->-\text{infinity}~\text{and}~\text{as}~x->+\text{infinity}.\]

Because fourth powers are positive for either sign of a large input, both polynomial ends approach negative infinity, choice A.

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Another way

  1. Compare p directly with the simpler parent model negative x to the fourth, which has both ends down.

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Common mistake

The constant negative 1 and linear term 2x do not control end behavior; always begin with the highest-power term.

Solution walkthrough video