Degree of a Polynomial Practice Test

Identify polynomial degree after simplification, multiplication, powers, composition, cancellation, and parameter changes.

Degree of a Polynomial Practice Test

20 varied questions on polynomial degree and leading terms.

Instant feedback · Worked explanations

Polynomial degree observatory

Find the highest power that survives.

Polynomial degree is not always the exponent that looks largest at first glance. The expression must be simplified first. Leading terms can combine, cancel, or disappear when a parameter takes a special value. Products, powers, composition, derivatives, and multivariable terms each have their own degree rules, so the safest strategy is to identify the structure before reading the final degree.

Anchor 01Simplify before naming the degree.

Anchor 02The leading nonzero term controls the degree.

Anchor 03Products add degrees; powers and composition multiply them.

Anchor 04The zero polynomial does not have ordinary degree zero.

1. Degree is the exponent on the surviving leading term

In standard form, a nonzero one-variable polynomial is arranged from highest power to lowest power. After like terms are combined, the first nonzero term is the leading term, and its exponent is the polynomial degree.

P (x) = a_{n} x^{n} + lower-degree terms

Write in standard formArrange terms from greatest power to least.

Combine like termsTerms with matching variable powers may change the apparent leader.

Remove cancellationsA leading term that becomes zero no longer determines degree.

Find the first nonzero termThis is the true leading term.

Read its exponentThat exponent is the degree.

Nonzero constant

Degree zero

\deg (c) = 0, c \neq 0

A nonzero constant has no visible variable factor, but it can be viewed as a constant times the zeroth power.

Leading coefficient

Coefficient is not degree

In $- 12 x^{7} + 3 x - 1$ , the leading coefficient is $- 12$ , while the degree is $7$ .

Illustrative example

Read after simplification

$4 x^{5} - 3 x^{2} + 1$ is already in standard form, so its degree is $5$ .

2. Cancellation can lower the degree

The most common trap is reading the largest exponent before simplifying. If two leading terms cancel, the degree falls to the next surviving power.

Cancellation gate

The apparent highest level disappears when its coefficient becomes zero.

Before simplification

5 x^{6} + 2 x^{3} - 5 x^{6} + x

Leading terms cancel

5 x^{6} - 5 x^{6} = 0

What survives

2 x^{3} + x

Final degree

3

Same degree, no cancellationIf leading coefficients add to a nonzero number, the top degree remains.

Same degree, full cancellationIf leading coefficients sum to zero, inspect the next power.

Different degreesThe higher-degree term normally survives because lower powers cannot cancel it.

Repeated simplificationMore than one level can disappear, so continue until the leading nonzero term is clear.

Key habit: do not decide the degree from the unsimplified appearance of a sum or difference.

3. Products, powers, sums, and composition change degree differently

The operation symbol tells you which degree rule applies. A valid rule for multiplication can be completely wrong for addition.

Product

\deg (P \cdot Q) = \deg (P) + \deg (Q)

For nonzero polynomials, add the factor degrees. Do not multiply them.

Power

\deg (P^{k}) = k \cdot \deg (P)

Raising a polynomial to a positive integer power repeats the degree contribution.

Composition

\deg (P \circ Q) = \deg (P) \cdot \deg (Q)

For nonconstant polynomials, composition multiplies degrees.

Sum

\deg (P + Q) \leq \max (\deg (P), \deg (Q))

The degree cannot exceed the larger input degree, and cancellation can make it smaller.

Illustrative product

Add factor degrees

If $\deg (P) = 3$ and $\deg (Q) = 5$ , then $\deg (P \cdot Q) = 3 + 5 = 8$ .

Illustrative composition

Multiply input degrees

If $\deg (P) = 4$ and $\deg (Q) = 2$ , then $\deg (P \circ Q) = 4 \cdot 2 = 8$ , provided both are nonconstant.

4. In several variables, add exponents within each term

A multivariable monomial has total degree equal to the sum of all exponents in that term. A multivariable polynomial takes the largest total degree among its nonzero terms.

One monomial: collect exponent contributions

$x^{3}$ contributes

3

$y^{2}$ contributes

2

$z$ contributes

1

degree (5 x^{3} y^{2} z) = 3 + 2 + 1 = 6

Polynomial: compare total term degrees

For $4 x^{2} y^{3} + 7 x^{5} - 2 y$ , the first term has total degree $2 + 3 = 5$ ; the second has degree $5$ ; the last has degree $1$ . Therefore the polynomial degree is $5$ .

Do not use only the largest single exponent. Total degree is computed term by term.

5. The zero polynomial is a special case

A nonzero constant has degree zero, but the zero polynomial is different because it has no leading nonzero term at all.

P (x) = 0

Nonzero constant

\deg (7) = 0

Zero polynomialIts degree is usually described as undefined in elementary algebra.

Why the distinction mattersCalling it degree

0

would make several degree rules inconsistent.

6. Parameters can switch the degree

A coefficient containing a parameter may become zero for a special parameter value. When that happens, the apparent leading term disappears and the degree drops.

(a - 2) x^{5} + 3 x^{3} - 1

Case 1: leading coefficient survives

If $a \neq 2$ , then $a - 2 \neq 0$ , so the degree is $5$ .

⇄

Case 2: leading coefficient vanishes

If $a = 2$ , then the fifth-degree term disappears and the degree becomes $3$ .

Parameter strategy: set the apparent leading coefficient equal to zero, solve that condition, then inspect the next surviving term.

7. Differentiation usually lowers degree by one

For a nonconstant polynomial, differentiation lowers the degree by one because the leading power loses one exponent while its new coefficient remains nonzero.

General rule

\deg (P^{'}) = \deg (P) - 1

This assumes a nonconstant polynomial. A constant differentiates to the zero polynomial, which must be handled separately.

Illustrative example

If $P (x) = 5 x^{6} - 2 x^{2} + 1$ , then $P^{'} (x) = 30 x^{5} - 4 x$ . The degree changes from $6$ to $5$ .

8. Check that the expression is actually a polynomial

Degree rules for polynomials apply only when variable exponents are nonnegative integers. Negative exponents or variables in denominators move the expression outside ordinary polynomial form.

Valid polynomial

Allowed exponents

$3 x^{4} - \frac{1}{2} x^{2} + 7$ is a polynomial. Fractional coefficients are allowed because the restriction is on variable exponents.

Not a polynomial

Negative variable exponent

$x^{- 2} + 4$ is not a polynomial in $x$ because the exponent is negative.

9. Common degree errors

Most mistakes come from choosing a familiar rule before checking what operation or simplification is actually present.

Leading coefficient used as degree

In $9 x^{4}$ , the coefficient is $9$ ; the degree is $4$ .

Exponents added across a sum

Addition does not add degrees. The largest surviving power controls the result.

Cancellation ignored

Leading terms can disappear completely, forcing the degree downward.

Factor degrees multiplied

For a product, factor degrees are added, not multiplied.

Zero polynomial called degree zero

A nonzero constant has degree zero; the zero polynomial is a special undefined case in elementary algebra.

Negative exponent treated as polynomial degree

An expression with a negative variable exponent is not an ordinary polynomial in that variable.

10. Degree rule shelf

These compact rules are useful only after the expression has been identified correctly.

Nonzero constant

\deg (c) = 0, c \neq 0

Degree zero.

Product

\deg (P \cdot Q) = \deg (P) + \deg (Q)

Add factor degrees.

Power

\deg (P^{k}) = k \cdot \deg (P)

Multiply degree by the outside exponent.

Composition

\deg (P \circ Q) = \deg (P) \cdot \deg (Q)

Multiply degrees for nonconstant polynomials.

Sum

\deg (P + Q) \leq \max (\deg (P), \deg (Q))

Cancellation can reduce the maximum.

Derivative

\deg (P^{'}) = \deg (P) - 1

Usually lowers degree by one.

Multivariable term

degree = sum of exponents

Add exponents within one term.

Zero polynomial

P (x) = 0

No ordinary finite degree in elementary algebra.

11. Worked identification mini-set

These are independent illustrative examples, not questions copied from the test.

Example A

Cancellation

$2 x^{5} - 2 x^{5} + 7 x^{2}$ simplifies to $7 x^{2}$ , so the degree is $2$ .

Example B

Product

A degree- $3$ polynomial times a degree- $4$ polynomial has degree $7$ , assuming both factors are nonzero.

Example C

Power

If $\deg (P) = 2$ , then $\deg (P^{5}) = 5 \cdot 2 = 10$ .

Example D

Composition

A degree- $3$ outer polynomial composed with a degree- $2$ inner polynomial has degree $6$ .

Example E

Parameter switch

In $(a + 1) x^{4} + x^{2}$ , the degree drops below $4$ when $a = - 1$ .

Example F

Derivative

A nonconstant degree- $9$ polynomial has a first derivative of degree $8$ .

Final degree checklist

Before choosing an answer, check whether the apparent highest power really survives all simplification and special conditions.

Did I simplify first?Combine like terms and remove cancellations before reading the leading term.

Is the expression a polynomial?Negative or fractional variable exponents invalidate ordinary polynomial degree rules.

Which operation is present?Product, power, composition, sum, and derivative use different degree rules.

Could a parameter kill the leading coefficient?If so, solve that condition and inspect the next surviving term.

For several variables, did I add exponents within each term?Compare total term degrees, not just individual exponents.

Is this the zero polynomial?Do not automatically label it degree zero.