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Introduction to Polynomials0%

Polynomials & Quadratics · Topic 1 of 10

Introduction to Polynomials

Video lesson · from 16:57

One lesson video covers all of Polynomials & Quadratics, so it opens at 16:57 for this topic — not from the beginning.

Theory

Polynomials are expressions with one or more terms of the form:

anxn+an−1xn−1+an−2xn−2+...+a2x2+a1x1+a0x0a_n x^n+a_{n-1} x^{n-1}+a_{n-2} x^{n-2} + ... + a_2 x^2+a_1 x^1+a_0 x^0

where a0,…,ana_0, \dots, a_n are constants with an≠0a_n \neq 0.

Each term has a coefficient followed by a variable raised to a whole number exponent.

For example,

  • 3x5+x3+2x2−63x^5+x^3+2x^2-6 is a polynomial of degree 5 as the highest power of xx is 5.
  • Quadratics are polynomials of degree 2.
  • Constants are polynomials of degree 0 since 1=x01 = x^0.

Task: Complete the table

PolynomialDegree
4x2+2x+54x^2+2x+52
4x4−x3+x2−x+14x^4-x^3+x^2-x+14
3x23x^22
2x+12x+11
(x−1)4(x-1)^44
3x2+2x53x^2+2x^55
550

⚠️ Common Examiner Traps

  • The degree is the highest power: not the number of terms, and not the first coefficient you see.
  • Write terms in descending order: before doing anything else, and include a zero for any missing power — a missing term will wreck synthetic division.
  • A root is where the graph crosses: a root x=ax = a corresponds to a factor (x−a)(x-a), with the sign reversed.
  • Degree tells you how many roots to expect: a cubic has at most three, counting repeats.