Polynomial Functions Math Example 1

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Example 1

easy
Find the degree and leading coefficient of p(x)=−3x4+7x2−x+5p(x) = -3x^4 + 7x^2 - x + 5.

Solution

  1. 1
    Write the polynomial in descending powers and identify the highest exponent present.
  2. 2
    The highest power is x4x^4, so the degree is 44.
  3. 3
    The coefficient attached to the leading term −3x4-3x^4 is −3-3, so the leading coefficient is −3-3.

Answer

Degree 4,leading coefficient −3\text{Degree } 4, \quad \text{leading coefficient } -3
The degree determines the end behavior and maximum number of roots. A negative leading coefficient with even degree means the graph falls on both ends.

About Polynomial Functions

A polynomial function is formed by adding terms of the form axnax^n where nn is a non-negative integer. The highest power determines the degree, which controls the graph's end behavior, maximum turning points, and number of possible real zeros.

Learn more about Polynomial Functions →

More Polynomial Functions Examples

Example 2 medium

Find all zeros of [formula].

Example 3 medium

Perform polynomial long division: [formula].

Example 4 hard

Sketch the end behavior and find all real zeros of [formula]. State the multiplicity of each zero.

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