Title:-
Different types of polynomials:-
Introduction:-
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Today, in this article I will try to explain the topic Different types of polynomials. Hope this will help on going school students a lot.
Explanation:-
What is a polynomial?
Answer:-
In Mathematics, a polynomial is an expression which consists one or more terms and co-efficients that take part only the operations of addition, subtraction, multiplication, and positive integer power of variables and integer power will never be a negative one or a fraction,or never be the square root of variables .e.g, x² - x - 6 is a quadratic polynomial of variable x as the degree of the polynomial is 2 (in terms of degree if we consider) again it can be called as trinomial (if we consider its total terms).
How polynomials can be characterised?
Answer:- Depending on terms polynomials are characterised in four types
and they are :-
(i) Monomial
(ii) Binomial
(iii) Trinomial
(iv)Quadronomial.
Again depending on degree polynomials are further divided in four types and they are:-
(i) Zero degree polynomial.
(ii) Linear or one degree polynomial.
(ii) Quadratic or two degree polynomial.
(iii) Cubic or three degree polynomial.
What is monomial?
Answer :- The polynomial which contains only one term or a single term is known as monomial.e.g; 2x, 10……..etc.
What is binomial ?
Answer:- The polynomial which hasThe polynomial which has two terms is known as binomial. e.g; x + y, 2x -5….etc
What is trinomial?
Answer:- A polynomial which has three terms is known as trinomial.e.g; x² +y³ -7,
2x + y + 5
What is quadronomial?
Answer:- A polynomial which has four terms is known as quadronomial.e.g; 3y³+ x² - z + p, 5z -3y + x -9 …. etc.
What is a zero degree polynomial?
Answer:- A polynomial whose highest power of variables is zero is called a zero degree polynomial.e.g; 2x⁰, x⁰ + 5y⁰ + z⁰ ......etc.
What is one degree or linear polynomial?
Answer:- The polynomial whose highest exponent of variable is one is called one degreeThe polynomial whose highest exponent of variable is one is called one degree or linear polynomial.e.g; x + y , 9x + 3y +8 are linear polynomials of two variables…..etc.
What is two degree or quadratic polynomial?
Answer:- The polynomial whose highest power of variable or sum of exponents of variables is 2 is known as two degree or quadratic polynomial. e.g; 2x² + x + 3 is a quadratic polynomial of single variable x .
again,
xy + 5z + 2 is a quadratic polynomial of three variables x,y and z. ∵ The powers of variables of x and y are 1and 1,
Now sum of the powers of variables x and y is = 1+ 1 = 2 , so it is a quadratic polynomial.
What is Cubic polynomial?
Answer:- A polynomial whose highest variable's power is three is called a cubic polynomial. e.g; x³ + x² +x + 2 is a cubic polynomial of single variable x
5x²y + xy + z is a cubic polynomial of thre variables x,y and z.
Now let, us do some problems of polynomials:-
Find the degree of the following polynomials:-
(i) 4x² + x + 9 (ii) x³ y² + 7y + m
(iii) x - 3y + 5z
Answer:- (i) The degree of this poly nomial = 2
Explanation:-As,its highest power of variable x is = 2
(ii) The degree of this polynomial = 3 + 2 = 5
Explanation:- As the highest power of variable x = 3 and highest power of variable y = 3
So its degree is = 3 + 2 = 5
(iii) The degree of this polynomial = 1
Explanation :- The highest exponents of the variables x, y and z are equal and it is 1,
So, degree of this polynomial is = 1
Find out which of the following expressions are polynomials also write their name.
(i) 1/x + y -7 (ii) y²/³ + 9x + 4 (iii) x³ + y² + z
Answer:- (i) It is not a polynomial.
Explanation:- The exponent of variable x is = - 1 which is a negative integer.
So, it is not a polynomial.
(ii) It is not a polynomial.
Explanation:- The exponent of variable y is = ⅔ which is a fraction, so it is not a polynomial.
(iii) It is a cubic polynomial ( If we consider its degree)
It is also a trinomial.(If we consider its terms)
Explanation:-
If we consider degree,
Its highest exponent of variable x is = 3 , so it is a cubic polynomial.
Again if we consider terms,
The total number of terms of the expression is = 3
So, it is a trinomial.
Find a quadratic polynomial
whose roots are (i) - 2 , 3 (ii) ¼, 4 (iii) 2, 2
Answer :-
(i) ∵ sum of the roots(s) = - 2 + 3 = 1
and product of the roots (p) = - 2 × 3
= - 6
∴ Quadratic polynomial will be,
x² - sx + p
= x² - 1.x + (- 6)
= x² - x - 6
(ii) ∵ sum of the roots (s) = ¼ + 4/1
= (1 + 16)/4 [ ∵ L.C.M of 1, 4 is 4]
= 17 /4
product of the roots(p) = ¼ × 4
= 1
∴ quadratic equation is,
x² - sx + p = 0
=> x² - 17/4x + 4 = 0
=> 4(x² - 17/4x + 4) = 0 [ multiplying both sides by 4 ]
=> 4x² - 17x + 16 = 0
∴ quadratic polynomial is = 4x² - 17x + 16
(iii)
FREQUENTLY ASKED QUESTIONS:-
What are the four types of polynomial?
Answer :- The four types of polynomials are :-
(i) Monomial:- The polynomial which contains only one term or a single term is known as monomial.e.g; 2x, 10……..etc.
(ii) Binomial :- The polynomial which has two terms is known as binomial. e.g; x + y, 2x -5….etc
( iii) Trinomial :- A polynomial which has three terms is known as trinomial.e.g; x² +y³ -7,
2x + y + 5
(iv) Quadronomial:-A polynomial which has four terms is known as quadronomial.e.g; 3y³+ x² - z + p, 5z -3y + x -9 …. etc
What is a 6th degree polynomial called?
Answer :- A 6th degree polynomial is called a sextic polynomial.It is also known as hexic polynomial
What is a zero degree polynomial?
Answer:- A polynomial whose highest power of variables is zero is called a zero degree polynomial.e.g; 2x⁰, x⁰ + 5y⁰ + z⁰ ......etc.
Is a number a polynomial?
Answer :- Yes, any number is always a polynomial and it is a zero degree polynomial. e.g; 2 is a zero degree polynomial. we can express 2 as,
2 = 2 × 1 = 2.x° = 2x°
So 2 can be called as zero degree polynomial.
Which one is a polynomial?
(a)1/(x+y) (b)3x + y -3
(c)√(4m² + 4n²) (d) x⁻³ + 2y² + 3
Answer :- (b)3x + y -3
What are the four types of polynomial?
Answer :- The four types of polynomials are :- (i) Monomial:- The polynomial which contains only one term or a single term is known as monomial.e.g; 2x, 10……..etc. (ii) Binomial :- The polynomial which has two terms is known as binomial. e.g; x + y, 2x -5….etc ( iii) Trinomial :- A polynomial which has three terms is known as trinomial.e.g; x² +y³ -7, 2x + y + 5 (iv) Quadronomial:-A polynomial which has four terms is known as quadronomial.e.g; 3y³+ x² - z + p, 5z -3y + x -9 …. etc
What is a 6th degree polynomial called?
Answer :- A 6th degree polynomial is called a sextic polynomial.It is also known as hexic polynomial
What is a zero degree polynomial?
Answer:- A polynomial whose highest power of variables is zero is called a zero degree polynomial.e.g; 2x⁰, x⁰ + 5y⁰ + z⁰ ......etc.
Is a number a polynomial?
Answer :- Yes, any number is always a polynomial and it is a zero degree polynomial. e.g; 2 is a zero degree polynomial. we can express 2 as, 2 = 2 × 1 = 2.x° = 2x° So 2 can be called as zero degree polynomial.
Which one is a polynomial? (a)1/(x+y) (b)3x + y -3 (c)√(4m² + 4n²) (d) x⁻³ + 2y² + 3
Answer :- (b)3x + y -3
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