L2 · Classifying Polynomials: Terms and Degree; Conventional Order

Learning goal

“I can classify a polynomial by its number of terms and by its degree, name terms by degree, justify whether an expression is a polynomial, and rewrite a polynomial in conventional descending (or ascending) order of degree.”

Polynomial — a sum of monomials whose coefficients are real numbers and whose variable exponents are all natural numbers (1, 2, 3, ...). A constant term counts as degree zero.

Degree of a term — the exponent value in the term: x² has degree 2, x has degree 1, a constant has degree 0.

Degree of a polynomial — the greatest degree of any of its terms. Polynomials are named by degree: 1 linear, 2 quadratic, 3 cubic.

Like terms — terms with the same variable and the same degree, like 3x² and -5x². (They become the working tool of L3.)

Worked example

Take the polynomial 5x³ - 2x + 7. Step 1: Sort it into terms - the chunks separated by + and - signs: 5x³, -2x, +7. Three terms - so it is a trinomial. Step 2: Read each term's degree: 5x³ has degree 3; -2x has degree 1 (x is x¹); 7 is a constant, degree 0. Step 3: The degree of the polynomial is the GREATEST term degree: 3. A degree-3 polynomial is named cubic. Say it plainly: number of terms answers "how many chunks?"; degree answers "what is the highest exponent anywhere?" - two different questions, two different labels, and every polynomial carries both.

Check yourself — explain it

Why is the constant 7 assigned degree 0?

Paired problem — example still visible

Classify 2x² + 5 by number of terms, and give its degree -> Binomial (two terms: 2x² and 5); degree 2 (quadratic)

Faded completion — last step is yours

"4x⁵ - x² + 9 has term degrees 5, 2 and 0, so the degree of the polynomial is ___" -> 5

Ready to practise?

10 scored questions with a 3-rung hint ladder. Mastery needs 8/10 plus both gate questions.

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