Polynominals

Polynominals

Understanding Polynomials

  • A polynomial is an expression consisting of variables and coefficients, with operations being addition, subtraction, multiplication, and non-negative integer exponentiation of variables.
  • Degree of a polynomial is the highest power of the variable in a polynomial. For instance, in the polynomial 4x^3+2x^2+3, the degree is 3.

Types of Polynomials

  • Polynomials can be monomial (e.g., 3x), binomial (e.g., 3x+2), or trinomial (e.g., 3x^2+2x+1) based on the number of terms in them.
  • A polynomial of degree 1 is known as a Linear Polynomial, of degree 2 as a Quadratic Polynomial, and of degree 3 as a Cubic Polynomial.

Operations on Polynomials

  • Addition and Subtraction of polynomials involve combining or subtracting the like terms.
  • Multiplication of polynomials is carried out using the distributive property.
  • Division of polynomials can be done through long division or synthetic division methods.

Polynomial Equations and Roots

  • A polynomial equation is an equation in which a polynomial is set equal to another polynomial or a constant.
  • Roots of a polynomial are the values of the variable that make the polynomial zero.

The Remainder Theorem and The Factor Theorem

  • The Remainder Theorem states that when a polynomial f(x) is divided by (x-a), the remainder is f(a).
  • The Factor Theorem states that a polynomial f(x) has a factor (x-a) if and only if f(a)=0.

Polynomial Graphs

  • The graph of a polynomial function of degree n will have at most (n-1) turning points.
  • The end behaviour of a polynomial refers to the behaviour of the graph as x approaches positive or negative infinity.

Properties of Polynomials

  • Polynomials are infinitely differentiable.
  • The Fundamental Theorem of Algebra states that every non-zero single-variable polynomial with complex coefficients has at least one complex root.

These are the major topics you should focus on while revising polynomials. This knowledge should give you good baseline knowledge to tackle any polynomial question you might face in the exam.