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Degree and Leading Coefficient Practice Problems

Finding the degree and leading coefficient is quick when a polynomial is tidy and easy to get wrong when it is not. These twelve problems start with polynomials in descending order and work up to scrambled, cancelling, and factored forms where the leading term has to be found rather than read.

  1. 1.

    What is the degree of ?

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    The polynomial is in descending order, so the first term is the leading term. Its exponent, 4, is the degree.

  2. 2.

    What is the leading coefficient of ?

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    The leading term is and its coefficient, sign included, is . Dropping the minus sign is the most common slip here.

  3. 3.

    What is the constant term of ?

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    The constant term is the one with no variable: . It is also the -intercept of the graph, since .

  4. 4.

    What is the degree of the constant polynomial ?

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    A non-zero constant is , so its degree is 0. Only the zero polynomial has an undefined degree.

  5. 5.

    What is the degree of ?

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    The polynomial is out of order. Scan the exponents — 1, 6, 2, 0 — and take the largest. The degree is 6, from the term, not 1 from the term written first.

  6. 6.

    What is the leading coefficient of ?

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    The leading term is the one with the highest power, , so the leading coefficient is . In standard form the polynomial reads .

  7. 7.

    What is the leading coefficient of ?

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    The leading term is . An unwritten coefficient is 1, and the minus sign makes it . "There isn't one" is not an option — every non-zero polynomial has a leading coefficient.

  8. 8.

    After simplifying, what is the degree of ?

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    The two terms cancel: . What remains is , a quadratic, so the degree is 2. Always combine like terms before reading the degree.

  9. 9.

    Without expanding, what is the degree of ?

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    Add the degrees of the factors: . You never need to multiply the brackets out to find the degree of a product.

  10. 10.

    Without expanding, what is the leading coefficient of ?

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    Multiply the leading coefficients of the factors: . The expansion begins .

  11. 11.

    Without expanding, what is the leading coefficient of ?

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    The factor has leading term , so has leading coefficient . Multiply: . The two minus signs cancel and the leading coefficient is positive 3, with degree .

  12. 12.

    A polynomial of degree has 4 turning points and both tails going up. What is the smallest possible value of ?

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    Four turning points needs , so . But both tails going the same way requires an even degree, so 5 is ruled out and the smallest possibility is 6. Tails both *up* also tells you the leading coefficient is positive.
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MPhil in Mathematics, Founder of Mathovia
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Degree and Leading Coefficient Practice Problems

The degree and leading coefficient lesson comes down to finding one term — the one with the highest power — and reading two numbers off it. These twelve problems make that term progressively harder to find.

The two numbers

The degree is the largest exponent; the leading coefficient is the number in front of it, sign included; the constant term is .

When the polynomial is out of order

Rewrite in descending powers first, or at least scan every exponent:

Degree , leading coefficient . The first term written, , has nothing to do with either.

When the polynomial is factored

Add the factor degrees; multiply the factor leading coefficients:

The factor is the trap: its leading term is , which contributes a each time it appears.

The counts the degree controls

Both are ceilings. They are most useful in reverse — a graph with four turning points needs a degree of at least five.

Frequently Asked Questions

How do you find the degree of a polynomial?+

Simplify it, then take the largest exponent on the variable among all the terms. If the polynomial is written out of order, scan every term rather than reading the first one. If it is factored, add the degrees of the factors instead of expanding.

What is the leading coefficient?+

The number in front of the term with the highest power, including its sign. For −4x³ + x it is −4. An unwritten coefficient is 1, so for x⁵ − 2 the leading coefficient is 1.

How do I find the leading coefficient of a factored polynomial?+

Multiply the leading coefficients of each factor together, along with any constant out front. Watch for factors written backwards like (1 − x), whose leading coefficient is −1.

Is a constant a polynomial, and what is its degree?+

Yes. A non-zero constant such as 9 is a polynomial of degree 0, because 9 = 9x⁰. The zero polynomial is the single exception — its degree is undefined.

Why does the degree matter?+

It caps what the graph can do: a degree-n polynomial has at most n real zeros and at most n − 1 turning points, and whether n is even or odd decides whether the two tails of the graph point the same way.

What is not a polynomial?+

Anything with a negative or fractional exponent on the variable, a variable in a denominator, a variable under a root, or a variable in an exponent. The coefficients can be any real numbers — fractions and square roots as coefficients are fine.

More practice problems

End Behavior Practice Problems Graphing Polynomials Practice Problems Polynomial in Standard Form Practice Problems