Trinomials and its properties

Define trinomial or what is a trinomial?
A trinomial is a polynomial which consists of three monomials. For example: - 3x^2 + 4x - 8, 6m^2 + 4m+9 and 5x^3 + 3x^2 + 6x etc. all are the examples of trinomials.

Factoring trinomials
A trinomial exactly has three terms. Two trinomials always have same number of terms but their degree may differ. For example we may have two degree trinomial or three degree polynomial. A trinomial can be factored into its factors which when multiplied gives the same trinomial. A trinomial which is in a quadratic form that is with degree two can be factored using the quadratic formula. A polynomial with a degree greater than 2 can be first simplified by taking the common factors. If the trinomial is in a quadratic form such as ax^2 + bx +c, for factorization we follow the following steps:-
• Identify the values of a, b and c in the trinomial ax^2+ bx + c.
• Write down all the factors of ‘a*c’.
• Identify which factor pair sums to the value of ‘b’.

Adding and Subtracting trinomials

Adding trinomials
Trinomials can be added by simply adding the like terms together. For example: - The term with the same variables are added together. For example: - Add (3x^2 + 4x + 5) to (6x^3 + 9x^2 + 5)
= 3x^2 + 4x + 5 + 6x^3 + 9x^2 + 5
= 6x^3 + 12x^2 + 4x + 10

Subtracting trinomials
The terms of the trinomials can only be added or subtracted if the variables are same. The first step is to remove the brackets and put the like terms together that is the terms with same variables together and then perform the operations. For example: - (4x^2 + 7x - 2) – (2x^2 + 8x - 9)
= 4x^2 + 7x – 2 – 2x^2 - 8x + 9
= 4x^2 – 2x^2 + 7x – 8x – 2 + 9
= 2x^2 – x + 7
This is the resultant trinomial.
A trinomial, ax^2 + bx + c is called a perfect square trinomial if b^2 = 4ac
A perfect square trinomial is obtained when we multiply a binomial by itself; the resulting polynomial is known as Perfect square trinomial. For example : - (x-2) is a binomial. If we multiply (x-2) by itself we will get (x-2) (x-2) which is equal to x^2 + 4 - 4x which is a perfect square trinomial.

Fun with Fractions

Fractions:
A part of a whole is called a fraction.
Nickel = 5 cents
Dime = 10 cents
Dollar = 100 cents
All are parts of a dollar. Hence, they can be represented as follows.
Nickel = one twentieth of a dollar (1/20).
Dime = One tenth of a dollar (1/10).
Cent = one hundredth of a dollar (1/100).
We can exchange 2 nickels for a dime. It means that the monetary value of 2 nickels and a dime are equal. When we represent them as fractions, we get
2 nickels = 2 X 1/20 = 2/20
1 Dime = 1/10
Thus 2/20 = 1/10. Even though their numerators and denominators are different.

What is Equivalent Fractions?
Fractions, having same value but with different numerators and denominators are called equivalent fractions.
Example: 1/3 and 3/9
Though the fractions look different, their values are same. If we divide both numerator and denominator of 3/9 by 3, we get the same fraction of 1/3.
Similarly, if we multiply both numerator and denominator of 1/3 by 3, we get 3/9 which is equivalent t0 the other fraction.

Fractions Equivalent
Fractions, which are Equivalent, have two properties.
(1) Fractions will have same value.
(2) The numerator and denominator of one of the fractions will have a common factor.
Example:
1/3 = 1 x 3/3 x 3
The numerator and denominator of 3/9 can be reduced, as both are factors of 3.

Equivalent fractions Definition:
Fractions with different numerators and denominators having same value are called Equivalent fractions.

 Equivalent Fractions:
Let is learn how to find the equivalent fractions. There is a simple method of finding equivalent fractions.  They can be obtained by either dividing or multiplying both numerator and denominator of a fraction by the same number.
Example:
6/8= 6 ? 2/ 8 ? 2 =3/4. Hence, ¾ and 6/8 are equivalent fractions.
6/7 = 6 x 9/ 7 x 9 = 54/63. Hence, 6/7 and 54/63 are equivalent fractions.
Checking for equivalent fractions:
Two fractions are equal if the product of numerator of one fraction with denominator of the other is equal to the product of the other two numbers.
Explanation: Two fractions A/B and C/D are equal if product of a and d is equal to product of b and c.
A / B = C / D if
A x D = B x C.

What is an equivalent fractions?
A fraction that has the same value but with different numerator and denominator is an equivalent fractions.

Introduction To Irrational Number

To  Understand what  is  an  Irrational number we have to understand what is a rational  number? Rational number are those numbers which can be expressed in the form of numerator / denominator. Every rational numbers can be expressed as decimal form or in non- terminating  repeating decimal form..

Definition of Irrational number: Irrational numbers are those numbers which cannot be expressed  in  the form of numerator / denominator. all non- terminating non- recurring decimals are irrational number.

What is a irrational number
According to definition of Irrational numbers : Any number that cannot be expressed in the form of p/q , where p , q are integers , q>0 , p and q have no common factor (except 1) is called an irrational number.

What are irrational number

• Negative of an  irrational number  is an irrational  number. 
• The sum of a rational number and an irrational number is considered as an irrational number.
• The product of a non-zero rational number and an irrational number is found to be irrational in nature.
• The sum, difference , product , and quotient of two irrational numbers need not to be an  irrational number  because the sum of the irrational numbers √3  and  -√3 is 0 which is a rational number, whereas the sum of irrational number √2 and √5 is an irrational number
• The difference of an irrational numbers √2 and  -√2 is zero which is a rational number but √5 and √ 6 difference is irrational  number.
• The product of √18 and √ 2 is 6, which is a rational number and √7 and √6 is √42 which is irrational.
• The quotient  of irrational number √12 and √ 3 is 2 , which is a rational number  and √12 and √4 is √3 is irrational number.

What is an irrational number

A  number is an irrational number  , if it has non-terminating  an non repeating decimal representation.

• Let  us  take some Irrational numbers examples 0.12112111211112………., 0.30003000030000003…………….. cannot be expressed  in the form of p/q. 
• The decimal representation of √2=1.414215…………………. is neither terminating nor repeating so , it is an irrational number.