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All courses Math III · A-SSE.1.a 14 of 55
Interpret terms, factors, and coefficients in polynomial and rational expressions.

Read leading coefficient, parity, and both polynomial ends

Problem
For p(x)=-3x⁴+2x-1, enter leading term, leading coefficient, degree, degree parity, coefficient sign, left-end selector, right-end selector, and dominant-term statement.
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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 leading term

\[-3x^4+2x-1\Rightarrow~\text{leading term }-3x^4\]

The highest-power term determines what the polynomial does for very large \(x\)-values.

02

Use the degree

\[x^4\Rightarrow~\text{even degree}\]

An even degree means the left end and right end go in the same direction.

03

Use the sign of the leading coefficient

\[-3<0\]

A negative leading coefficient makes both ends point downward for an even-degree polynomial.

04

State the interpretation

\[\text{both ends go downward}\]

So the leading coefficient \(-3\) controls both ends downward for this even-degree polynomial.

05

State the final answer

\[\text{leading}~\text{term}=-3x^4;~\text{leading}~\text{coefficient}=-3;~\text{degree}=4;~\text{parity}=\text{even};~\text{coefficient}~\text{sign}=\text{negative};~\text{left}~\text{end}=p(x)→-\text{infinity}~\text{as}~x→-\text{infinity};~\text{right}~\text{end}=p(x)→-\text{infinity}~\text{as}~x→+\text{infinity};~\text{dominance}=\text{large}-|x|~\text{behavior}~\text{follows}~-3x^4\]

The leading coefficient \(-3\) controls both ends downward for an even degree polynomial.

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Another way

  1. You can compare the polynomial to the simpler model \(y=-x^4\), since the lower-degree terms \(2x-1\) matter much less than \(-3x^4\) for large \(|x|\).

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Common mistake

A common mistake is to look at the constant term \(-1\) or the linear term \(2x\), but neither controls the end behavior the way the leading term does.