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Review of Polynomials

Definitions

Let's learn a few definitions.
 

Factor of the number
When two or more numbers are multiplied, each of the numbers is called a factor of the number.
For example, in the product 5 ∗ 11 = 55, 5 and 11 are factors of 55.
 

Coefficient
Each factor is the coefficient of the product of other factors.
 

For example, in a term 3xy

  • 3 is a coefficient of xy
  • x is a coefficient of 3y
  • y is a coefficient of 3x
  • xy is a coefficient of 3
 
Generally, the numerical part of a term is called the numerical term of its coefficient.

Thus in the term 3xy, 3 is the numerical coefficient.

Exponent
Sometimes, the products are written as powers.

For example, 4 ∗  4∗ 4 is written as 43

4 ∗ 4 is written as 42

a ∗  a ∗ a is written as a3

In a3, 3 is called the exponent or power and 'a' is called the base; the exponent 3 dictates the number of times the base 'a' occurs as a factor in the product.

Monomial
A monomial is a term that is either a number or a variable with positive integral index or an indicated product of a number and one or more variables.
 

Examples:

  • 7 is a monomial since it is a number.
  • p is a monomial since it is a variable.
  • 7p is a monomial since it is an indicated product of a number 7 and p.
  • 7pq is also a monomial since it is a product of 7 and the variable 'pq'. 3/4 x2 y3 is also a monomial.

Polynomial

A polynomial is an indicated sum of monomials.

 

Examples:

  • 3x + 7

  • (4/7)x2 + 6x - 8
 

Degree of polynomials
The degree of a polynomial is the greatest degree of its various terms.
 
Example:

2x + 3 has two terms, namely 2x and 3. The degree of 2x is 1. The degree of 3 is 0. The greatest of the two degrees is 1.
 

Addition properties of polynomials

  1. A and B are two polynomials. By adding them, we get (A+B), which is also a polynomial. Hence, the set of polynomials has the Closure Property.

  2. A + B = B + A (Commutative Property)

  3. (A + B) + C = A + (B + C) (Associative Property)

  4. The zero polynomial is the identity element under addition.

  5. If 'A' is a polynomial, its additive inverse is -A. Thus, every polynomial has an additive inverse.
 
Example:

If A = 3x3 + 4x2 - x - 1
   B = 4x3 -3 x2+ 4x + 5
Find A + B and B + A

A + B  = (3x3 + 4x2 - x - 1) + (4 x3 - 3 x2+ 4x + 5)
          = (3 + 4) x3 + (4 - 3) x2 + (- 1 + 4) x + (- 1 + 5)
          = 7x3 + x2+ 3x + 4

B + A = ( 4x3 - 3x2 + 4x + 5) + (3x3 + 4x2 - x - 1)
          = (4x3 + 3x3) + (- 3x2 + 4x2) + (4x - x) + 5 - 1
          = 7x3 + x2 + 3x + 4

It can be seen that A + B = B + A.

Try these questions

State the coefficients and degrees of the polynomials.

  1. 10x5

    2. Answer: Coefficient = 10; degree = 5

  2. - 2.51 x4

    3. Answer: Coefficient = -2.51; degree = 4

  3. – 8

    4. Answer: Coefficient = - 8; degree = 0

  4. √3x2

    5. Answer: Coefficient = √3; degree = 2


    Find the value of monomial when x = 3, 4.

  5. 2x2

    6. Answer: when x = 3
    2 ∗(3)2 = 2 ∗9 = 18
    when x = 4
    2 ∗(4)2 = 2 ∗16 = 32

    Find the values of the monomials when x = 2, 3, - 1.5

  6. 3x2

    7. Answer: When x = 2,
    the value of
    3x2 = 3 ∗(2)2 = 12

    When x = 3;
    the value of 3x2
    = 3 ∗(3)2= 27

    When x = - 1.5,
    the value of 3x2
    = 3 ∗(-1.5)2 = 6.75

  7. -1.2 x2

    8. Answer: When x = 2,
    the value of -1.2 x2
    = -1.2 ∗(2)2 = -4.8

    When x = 3,
    the value of -1.2 x2
    = -1.2 ∗(3)2= - 10.8

    When x = - 1. 5
    the value of -1. 2x2
    = - 1.2 ∗( - 1.5 )2 = - 2.7

  8. 1/2x3

    9. Answer: When x = 2,
    the value of 1/2 x3
    = 1/2 ∗2 ∗2∗2 = 4

    When x = 3 ;
    the value of 1/2 x3
    = 1/2 ∗3∗ 3 ∗3 = 13.5

    When x = -1.5 the value of 1/2 x3
    = 1/2 ∗-1.5 ∗-1.5 ∗-1.5
    = -1.6875

  9. 2x3

    10. Answer: When x = 2,
    then the value of 2x3
    = 2 ∗(2)3 = 2∗ 8 = 16

    When x = 3,
    then the value of 2x3
    = 2 ∗(3)3 = 2 ∗27 = 54

    When x = - 1.5,
    then the value of 2x3
    = 2 ∗(-1.5)3 = - 6.75

    Simplify

  10. - 3x2 + ( 6x2 ) - ( -0.5x2 ) + ( 1.5x2)

    11. Answer: = - 3x2 + 6x2 + 0.5x2 + 1.5x2
    = ( - 3 + 6 + 0.5 + 1.5 ) x2 = 5x2


  11. ( - 3x ) + ( - 4x ) - ( 4.5 ) x + ( 2.5x )

    12. Answer: = ( - 3 - 4 - 4.5 + 2.5 ) x = - 9x


  12. ( 3x ) + ( - 4x ) - ( - 3x ) + ( - 7x )

    13. Answer: = 3x - 4x + 3x - 7x
    = (3 - 4 + 3 - 7) x = - 5x


  13. ( - 5x2 ) + ( 5.2x2 ) + ( 1.5x2 ) - ( 0.7x2 )

    14. Answer: = ( - 5 + 5.2 + 1.5 - 0.7 ) x2
    = ( 6.7 - 5.7 ) x2 = (1) x2 = x2

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