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Determine the leading coefficient of a polynomial function

For a nonzero polynomial function, the leading coefficient is the numerical coefficient of the term with the greatest exponent of the variable when the polynomial is expressed in standard form; it may be positive, negative, fractional, decimal, or an understood 1, and it is distinct from the degree and leading term. For a factored expression, the leading coefficient is found by combining the leading coefficients of the factors, supporting interpretation of the polynomial’s end behavior; more advanced coefficient analysis is not included.

Detailed Explanation: Determine the leading coefficient of a polynomial function

To determine the leading coefficient of a nonzero polynomial:

  1. Write the polynomial in standard form, with exponents in descending order.
  2. Find the term with the greatest exponent.
  3. Identify the numerical factor multiplying that term. This number is the leading coefficient.

For example, consider

f(x)=4−7x3+12x5+2x2.f(x)=4-7x^3+\frac{1}{2}x^5+2x^2.

First, rewrite the terms in descending order of exponents:

f(x)=12x5−7x3+2x2+4.f(x)=\frac{1}{2}x^5-7x^3+2x^2+4.

The term with the greatest exponent is

12x5.\frac{1}{2}x^5.

Therefore:

  • The leading term is 12x5\frac{1}{2}x^5.
  • The degree is 55.
  • The leading coefficient is 12\boxed{\frac{1}{2}}.

The leading coefficient is the number attached to the term with the highest power of the variable. If no number is written, the coefficient is understood to be 11; if the term begins with a negative sign, the coefficient is negative.

Learn by doing: Determine the leading coefficient of a polynomial function

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    Rational Root Theorem - Leading Coefficient (In Order)

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