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Identify the degree and leading coefficient of a polynomial

For a nonzero polynomial in one variable, the degree is the greatest exponent with a nonzero coefficient, and the leading coefficient is the coefficient of the term containing that exponent; rewriting terms in descending powers makes both features clear, even when powers are missing or coefficients are negative or fractional. Nonzero constants have degree zero, while the zero polynomial has no defined degree in this treatment; these concepts support polynomial comparison, graph behavior, and later factoring.

Detailed Explanation: Identify the degree and leading coefficient of a polynomial

To identify the degree and leading coefficient of a polynomial:

  1. Rewrite the terms in descending order of powers.
  2. Find the term with the greatest exponent.
  3. The exponent is the degree.
  4. The coefficient of that term is the leading coefficient.

Example:

p(x)=2x2−34x5−x7+6p(x)=2x^2-\frac{3}{4}x^5-x^7+6

Step 1: Rewrite in descending powers.

The exponents are 77, 55, 22, and 00 (the constant 66 has exponent 00):

p(x)=−x7−34x5+2x2+6p(x)=-x^7-\frac{3}{4}x^5+2x^2+6

Step 2: Identify the greatest exponent.

The greatest exponent is 77, so the degree is

7\boxed{7}

Step 3: Identify the coefficient of that term.

The term with x7x^7 is −x7-x^7. Since −x7=−1x7-x^7=-1x^7, its coefficient is −1-1.

Therefore, the leading coefficient is

−1\boxed{-1}

So, for this polynomial:

  • Degree: 7\boxed{7}
  • Leading coefficient: −1\boxed{-1}

Missing powers, such as x6x^6, do not affect the degree. A nonzero constant has degree 00, while the zero polynomial has no defined degree.

Learn by doing: Identify the degree and leading coefficient of a polynomial

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    Function End Behaviour (Polynomials) - Function to Power and Coefficient

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