Polynomials — degree, types (monomial, binomial, trinomial): Question bank

Before we start, one word we lean on constantly: a term is a single chunk of a polynomial — a number times a power of the variable, like $3 x^{2}$ or $- 5 x$ or the lone constant $7$ . Terms are separated by $+$ and $-$ signs. Everything below is about (a) counting terms after you simplify and (b) reading the degree = the highest exponent with a non-zero coefficient.

True or false — justify

x^{100} + 1

a binomial?

True. It has exactly two terms, so it is a binomial — the huge degree of

100

has nothing to do with the term count.

A trinomial always has degree

2

False. "Tri" counts terms, not degree.

x^{5} + x^{3} + x

is a trinomial of degree

5

; degree and term-count are independent.

The expression

2^{x} + 1

is a polynomial.

False. The variable sits in the exponent here, not in the base; polynomials only allow the variable raised to fixed whole-number powers like

x^{2}

, never

2^{x}

5

is a monomial.

True. Write it as

5 x^{0}

(since

x^{0} = 1

): one term, coefficient

5

, degree

0

— a perfectly valid constant monomial.

The polynomial

0

has degree

0

False. Every constant except zero has degree

0

; the zero polynomial has no non-zero term to measure, so its degree is undefined (often written

- \infty

x^{2} + 3 x - x^{2} + 1

is a trinomial.

False. Simplify first: the

x^{2}

terms cancel, leaving

3 x + 1

, which is a binomial. Always classify the simplified form.

x + 1

is a binomial polynomial.

False.

x = x^{1/2}

is a fractional exponent, which polynomials forbid, so it is not a polynomial at all — hence not a binomial.

A polynomial can have degree

0

True. Any non-zero constant, e.g.

- 7 = - 7 x^{0}

, is a degree-

0

polynomial (a constant monomial).

3 x + \frac{2}{x}

is a binomial.

False.

\frac{2}{x} = 2 x^{- 1}

is a negative exponent (division by the variable), which is not allowed, so the whole thing is not a polynomial.

Adding two binomials always gives a binomial.

False.

(x^{2} + x) + (x^{2} - x) = 2 x^{2}

— the

x

terms cancel, collapsing four terms into a single monomial.

Spot the error

5 x^{2} - 3 x^{2} + 2 x - 1

has four terms, so it is a polynomial with four terms."

The

x^{2}

terms combine:

5 x^{2} - 3 x^{2} = 2 x^{2}

, giving

2 x^{2} + 2 x - 1

, a trinomial. You must combine like terms before counting.

0 \cdot x^{5} + 3 x^{2} + 1

has degree

5

The leading coefficient is

0

, so that

x^{5}

term does not exist. The real expression is

3 x^{2} + 1

, degree

2

(x + 3) (x + 2) = x^{2} + 6

, a binomial."

FOIL gives

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

; the middle term

5 x

was dropped. The correct product is a trinomial.

x^{2} + 5 x + 6

is a monomial because it factors into

(x + 2) (x + 3)

Factored form is a product, not a single term. As written out,

x^{2} + 5 x + 6

has three terms, so it is a trinomial. Factoring does not change the type of the original expression.

"The constant

9

has no degree since there is no

x

9 = 9 x^{0}

, so its degree is

0

, not "none." Only the zero polynomial has undefined degree.

x^{- 2} + x^{- 1} + 1

is a trinomial of degree

- 1

Negative exponents are not permitted, so this is not a polynomial at all; talking about its "degree" or classifying it as a trinomial is meaningless.

"Since a binomial has

2

terms,

x^{2} - 4

must have degree

2

because it is a binomial."

The degree being

2

is a coincidence of this example, not a rule of binomials.

x^{100} - 4

is also a binomial but has degree

100

Why questions

Why do we forbid negative exponents like

x^{- 1}

in polynomials?

Because

x^{- 1} = \frac{1}{x}

is division by the variable, which is undefined at

x = 0

— polynomials are meant to be defined and smooth for every real number.

Why must we simplify before classifying a polynomial as mono/bi/trinomial?

The type counts distinct terms, and unsimplified like terms hide the true count —

x^{2} + 2 x^{2}

looks like two terms but is really one,

3 x^{2}

Why does the degree ignore all terms except the one with the highest power?

For large

∣ x ∣

the highest-power term grows fastest and dominates the value, so it dictates the polynomial's overall "power level" and end behaviour — the smaller terms become negligible.

Why is the leading coefficient required to be non-zero when we state "degree

= n

If the coefficient of

x^{n}

were

0

, that term contributes nothing and effectively disappears, so the true highest surviving power would be lower than

n

Why is "number of terms" a different idea from "degree"?

Term-count measures how many separate chunks are added together (structure), while degree measures the tallest exponent (height). One counts pieces, the other measures the biggest piece — they can vary completely independently.

Edge cases

Classify

7

: term-count and degree.

One term (monomial) with degree

0

, since

7 = 7 x^{0}

Classify the zero polynomial

P (x) = 0

It is a monomial-looking constant but a special one: its degree is undefined (or

- \infty

) because it has no non-zero coefficient anywhere to read a power from.

Can a monomial have degree

0

Yes — any non-zero constant such as

5 = 5 x^{0}

is a monomial of degree

0

a x^{m} + b x^{n}

a genuine binomial for all choices of

m

and

n

Only when

m \neq = n

and

a, b \neq = 0

. If

m = n

the terms combine into

(a + b) x^{m}

, becoming a monomial (or zero if

a + b = 0

What happens to the term-count when subtracting

x^{2} + x + 1

from

x^{2} + 2 x + 1

(x^{2} + 2 x + 1) - (x^{2} + x + 1) = x

, so three-term minus three-term collapses to a single-term monomial once like terms cancel.

Is the highest written exponent always the degree?

Not necessarily — only if its coefficient survives simplification and is non-zero. A written

x^{5}

with coefficient

0

, or one that cancels against another

x^{5}

, does not count toward the degree.

Recall One-line rule to carry away

Simplify first, then: count the surviving terms for the type, and read the highest surviving exponent for the degree — the two are independent, and only the zero polynomial has no degree at all.