Lesson 2: Polynomials. Algebraic Expressions

Key ideas: Polynomials. Algebraic Expressions

A term is a number, a variable, or a product of numbers and variables, such as 7, x or −4x²y. In −4x²y, the coefficient is −4 and the variables are x and y. A term with no variable, like 7, is a constant.

A polynomial is one term or a sum or difference of terms, where the variables have whole-number exponents (0, 1, 2, …). We name polynomials by the number of terms:

• Monomial: one term, e.g. 5m²
• Binomial: two terms, e.g. 3x − 7
• Trinomial: three terms, e.g. x² + 4x − 1
• The degree of a term is the sum of its variable exponents (x²y³ has degree 5). The degree of a polynomial is the highest degree of any of its terms.

Like terms have exactly the same variables raised to exactly the same exponents: 3x²y and −5x²y are like terms, but 3x²y and 3xy² are not. To add or subtract like terms, combine the coefficients and keep the variable part the same (4m + 12m = 16m). To multiply monomials, multiply the coefficients and add the exponents of matching variables (2m × 3m = 6m²).

Worked examples

Example 1

Classify each polynomial and state its degree: a) 4x³ − 2x + 7 b) −5a²b³

• a) Three terms, so a trinomial. The highest-degree term is 4x³: degree 3.
• b) One term, so a monomial. Degree = 2 + 3 = 5.

Example 2

Simplify 3x² + 5x − 4 − x² + 2x + 9.

• Group like terms: (3x² − x²) + (5x + 2x) + (−4 + 9).
• Answer: 2x² + 7x + 5

Example 3

Simplify (4y² − 3y + 1) − (2y² + 5y − 6).

• Subtracting a bracket changes the sign of every term inside it: 4y² − 3y + 1 − 2y² − 5y + 6.
• Collect like terms.
• Answer: 2y² − 8y + 7

Example 4

Multiply (−3a²b)(4ab³).

• Coefficients: (−3)(4) = −12.
• Variables: a² × a = a³ and b × b³ = b⁴.
• Answer: −12a³b⁴

Common mistakes

• Combining unlike terms: 3x + 2x² cannot be simplified to 5x³.
• Changing the exponent when adding: 4m + 12m = 16m, not 16m².
• Forgetting to change the sign of every term when subtracting a bracket.
• Forgetting the invisible coefficient: x means 1x, and −x means −1x.

Questions students ask

Is a single number such as 5 a polynomial?

Yes. It is a monomial (a constant term) with degree 0.

Is 3/x + 2 a polynomial?

No. 3/x = 3x⁻¹, and polynomials cannot have negative exponents on the variable.