POLYNOMIALS IN ONE VARIABLE

 

 

  • 2 Consider the polynomial 3x2 – 7x + 8
  • This is called a polynomial in one variable
  • 3x2, – 7x, 8 are called the terms.
  • 3 is the co – efficient of x2 and -7 is the co – efficient of x

 

2, -3, 7 etc are all examples of constant polynomials.

Polynomials having only one term are called monomials.

e.g. 3x, 2x3, -5x, etc

Polynomials having only two terms are called binomials.

e.g. x + 1, 3x2 – 7, etc

Polynomials having only 3 terms are called trinomials.

e.g. x2 + x – 3, 3x3 – 5x + 7 etc

The Highest power of the variable in a polynomial is called the degree of the polynomial               e.g. 7x5 – 3x2 +8

 

The highest power of the variable is 5. Hence the degree is 5

  • Power is also called exponent.
  • A polynomial of degree 1 is called linear polynomial
  • 2x – 1
  • A polynomial of degree 2 is called a quadratic polynomial
  • 3y + 6y2
  • A polynomial of degree 3 is called a cubic polynomial
  • 2x3 + 3x + 7
  • Maximum terms a polynomial can have
  • Linear polynomial Þ 2 terms
  • Quadratic polynomial Þ 3 terms
  • Cubic polynomial 4 terms
  • If a0 = a1 = a2 = a3 ….. = 0 we get the zero polynomial. The degree of the zero polynomial is not defined.

 

EXERCISE 2.1

 

  • Which of the following expression are polynomials in one variable and which are not? State the reason for your answer.

Solution:

  • Write the coefficients of x2 each of the following:

Solution:

  1. Co – efficient of x2 in 2 + x2 + x is 1
  2. Co – efficient of x2 in – x2 + x3 is -1
  • Co – efficient of x2 in x2 + x is
  1. Co – efficient of x2 in x – 1 is 0

 

3)      Give one example each of a binomial of (i) degree 35 (ii) degree 100.

Solution:

  1. One example of a binomial of degree 35 is 8x35 + 7x2
  2. One example of a monomial of degree 100 is 3x100

 

4)      Write the degree of each of the following polynomials:

Solution:

  1. Degree of polynomial 5x2 +  4x2 + 7x is 3
  2. Degree of polynomial 4 – y2 is 2
  • Degree of polynomial 5t – is 1
  1. Degree of polynomial 3 is 0

 

 

5)      Classify the following as linear, quadratic and cubic polynomials:

Solution:

(i)      Polynomial  x2  +  x Quadratic polynomial

(ii)     Polynomial  x  –  x3 Cubic polynomial

(iii)    Polynomial  y  +  y2 + 4 Quadratic polynomial

(iv)    Polynomial  1  +  x Linear polynomial

(v)     Polynomial  3t  Linear polynomial

(vi)    Polynomial  r2  Quadratic polynomial

(vii)   Polynomial 7 x3 Cubic polynomial