Introduction to Polynomials

The building blocks of algebra: variables, constants, and exponents.

What is a Polynomial?

A polynomial is an algebraic expression consisting of variables, constants, and exponents, combined using addition, subtraction, and multiplication.

Key Rule: The exponents of the variables must be whole numbers (0, 1, 2, ...).

Examples: 2x, x² + 2x, x³ - 5x + 3.
NOT Polynomials: 1/x (exponent -1), √x (exponent 1/2).

Terms and Coefficients

Terms: Parts of the polynomial separated by + or - signs.
Example: In 4x² - 3x + 7, the terms are 4x², -3x, and 7.

Coefficients: The numerical factor of a term.
Example: In 4x² - 3x + 7, coefficient of x² is 4, coefficient of x is -3.

Important Types

  • Constant Polynomial: A polynomial containing only a constant term (e.g., 5, -3, 7). Its degree is 0.
  • Zero Polynomial: The constant polynomial 0 is called the zero polynomial. Its degree is undefined.

Practice Questions

Free Preview - 10 Questions

Test your understanding of Polynomial basics.

1 Which of the following is a polynomial?
  • A x² + 1/x
  • B 2x² + 3√x
  • C x² + 3x + 4
  • D x⁻¹ + 5
Explanation:
In x² + 3x + 4, all exponents of x (2, 1, 0) are whole numbers. In others, exponents include negative numbers or fractions.
2 What is the coefficient of x² in the polynomial 5x³ - 4x² + 7x - 2?
  • A 5
  • B 4
  • C -4
  • D -2
Explanation:
The term with x² is -4x². The coefficient includes the sign, so it is -4.
3 The number of terms in the polynomial 3x² + 5x is:
  • A 1
  • B 2
  • C 3
  • D 0
Explanation:
The terms are 3x² and 5x. There are 2 terms. Such a polynomial is called a binomial.
4 Which of these is a constant polynomial?
  • A x
  • B x + 1
  • C 7
  • D 0x
Explanation:
A polynomial with only a constant term (and no variables with positive degrees) is a constant polynomial. 7 is a constant polynomial.
5 The exponent of a variable in a polynomial must be a:
  • A Real number
  • B Integer
  • C Whole number
  • D Rational number
Explanation:
By definition, the exponents of variables in a polynomial must be non-negative integers (whole numbers).
6 Is '0' a polynomial?
  • A Yes, it is called zero polynomial
  • B No, it is just a number
  • C It is a variable
  • D None of the above
Explanation:
The number 0 is a special polynomial called the Zero Polynomial.
7 Which is a trinomial?
  • A x + y
  • B x²
  • C x² + x + 1
  • D x³ + 2
Explanation:
A trinomial is a polynomial with exactly three terms. x² + x + 1 fits this description.
8 What is the coefficient of x in 2x² - x + 5?
  • A 2
  • B 1
  • C -1
  • D 0
Explanation:
The term is -x, which is equivalent to -1x. So the coefficient is -1.
9 Degree of the zero polynomial is:
  • A 0
  • B 1
  • C Any natural number
  • D Not defined
Explanation:
The degree of the zero polynomial is not defined.
10 Which is NOT a polynomial in one variable?
  • A 4x² - 3x + 7
  • B y³ + 1
  • C 3t + √2
  • D x + y
Explanation:
x + y has two variables (x and y). The others have only one variable (x, y, or t).

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