Polynomials

• To recognise a polynomial among other algebraic expressions — and explain why √x or 1/x fail the definition.
• To use precise vocabulary: term, coefficient, degree, monomial, binomial, trinomial, standard form.
• To add, subtract, and multiply polynomials by treating like terms as objects you can combine.
• To evaluate a polynomial at a given value and connect polynomials to area, cost, and pattern problems.

1. Definition — the three conditions

An expression in one variable x is a polynomial only if:

1. Powers are whole numbers — 0, 1, 2, 3, … (no negative or fractional exponents).
2. Coefficients are real numbers (integers, fractions, decimals are fine).
3. It can be written as a finite sum of terms.

2. Anatomy of a term

In 7x⁴: coefficient = 7, variable part = x⁴, exponent = 4.
In −3x: coefficient = −3, exponent = 1.
In 9: coefficient = 9, exponent = 0 (since 9 = 9x⁰).

Standard form: Write terms in descending order of degree:
aₙxⁿ + aₙ₋₁xⁿ⁻¹ + … + a₁x + a₀

3. Degree and naming

• Degree = highest exponent with non-zero coefficient.
  Example: 5x³ + 2x has degree 3 (the x term does not increase the degree).
• Monomial — 1 term (4x²). Binomial — 2 terms (x + 3). Trinomial — 3 terms (x² + 2x + 1).

4. Like vs unlike terms — the combining rule

Like terms share the same variable part (same variable(s) AND same power(s)).
2x² and 5x² are like → 2x² + 5x² = 7x².
x² and x are unlike → cannot combine to 2x² or 2x³.

Visual model: Think of x² tiles and x tiles as different shapes — only identical shapes merge.

5. Operations

Addition / subtraction: Align like terms; add/subtract coefficients only.
(3x² + 2x − 1) + (x² − 5x + 4) = 4x² − 3x + 3

Subtraction trap: (3x − 5) − (x − 2) = 3x − 5 − x + 2 = 2x − 3
(Change sign of every term in the subtracted polynomial.)

Multiplication: Distributive law.
x(2x + 3) = 2x² + 3x
(x + 2)(x + 3) = x² + 3x + 2x + 6 = x² + 5x + 6

Degree of a product: deg(p × q) = deg(p) + deg(q).
(x² + 1)(x³ − 2x) has degree 2 + 3 = 5.

6. Evaluating p(x) at x = a

Substitute a for x everywhere.
If p(x) = x³ − 2x² + x − 3, then p(1) = 1 − 2 + 1 − 3 = −3 and p(0) = −3.

A rectangular plot has length (2x + 5) m and breadth (x + 3) m. Write the area as a polynomial and find the area when x = 10.

Step 1 — Area = length × breadth = (2x + 5)(x + 3)
Step 2 — Expand: 2x·x + 2x·3 + 5·x + 5·3 = 2x² + 6x + 5x + 15
Step 3 — Combine like terms: 2x² + 11x + 15
Step 4 — At x = 10: 2(100) + 11(10) + 15 = 200 + 110 + 15 = 325 m²
Answer: Area polynomial = 2x² + 11x + 15; at x = 10, area = 325 m²

Add (2x² − 3x + 5) and (x² + 4x − 1).

x² terms: 2 + 1 = 3x²
x terms: −3 + 4 = x
constants: 5 + (−1) = 4
Result: 3x² + x + 4

• Classify by number of terms: 7x⁴, x + 1, 3x² + 2x − 5.
• Degree of 4x³ − 7x + 2? Degree of constant 100?
• Expand: (x + 4)(x − 2).
• If p(x) = x² − 3x + 2, find p(2) and p(0).
• Stretch: Is (x + 1)(x + 2)(x + 3) a polynomial? What is its degree?

Open the Practice tab for graded questions on Polynomials.