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Polynomials I Class 10 Math's I Chapter 2



Polynomials are algebraic expressions with one or more terms. These terms can are associated with mathematical operator such as addition, multiplication, subtraction or division. 

Polynomial word contains two terms poly- (meaning "many") and -nominal (in this case meaning "term") ... so it says "many terms".

Any term of polynomials do not contain the following:
• Division by a variable
• Negative exponents
• Fractional exponents
• Radicals (irrational number)

If you add or subtract polynomials, you get another polynomial. If you multiply them, you get another polynomial.

Polynomials often represent a function. And if you graph a polynomial of a single variable, you'll get a smooth, curvy line with continuity.

Examples of polynomial                                                   

• 7x (monomial)

• 3x − 2 (Binomial)

• −6y2 − (79) x (Binomial)

• 4xyz + 3xy2z − 0.1xz − 200y + 0.5 (Polynomial with 5 terms)

• 512v5 + 99w5 (Binomial)

• 5 (monomial)

• 5/x (not a polynomial)

• 3xy-2 (not a polynomial)

Based on number of terms in polynomials are categorized as

1. Polynomial with one term is called monomial.

2. Polynomials which contain two terms are called binomial.

3. Polynomials which contain three terms are called trinomial.

Polynomials with four or more terms are not given any special name. They are generally termed as polynomials.

There are three components or terms of any polynomial. These are usually

• Variables - These are letters like x, y, and m etc.

• Constants - These are numbers like 3, 5, 7 etc. They are sometimes attached to variables but are also found astern independently.

Exponents - Exponents are usually attached to variables but can also be found with a constant. Exponent in polynomial is always positive.
Examples of exponents include the 2 in 5² or the 3 in x³.

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Degree of a polynomial

Degree of polynomial is determined by identifying the degree of terms.

Case 1: Polynomial contains one variable:

The largest exponent in any term of the polynomial of one variable is known degree of polynomial.

The term containing largest exponent on variable is called leading term.

Case 2: Polynomial contains more than one variables:

1. Add the exponents of each variable in each term.

2. Maximum value of sum of exponents in any term of polynomial is the degree of polynomial.

3. The term containing largest exponent on variable is called leading term.

Polynomials are commonly written with their terms in descending order of degree.

The term with the highest degree is called the leading term.

Liner polynomial :

A polynomial of degree 1 is called liner polynomial. General form of linear polynomial p(x)=ax+b ,Where a& b are real number and a ≠0.Linear polynomials are the simplest form of polynomials.
Ex: p(x)= 4x + 5 is an example of linear polynomial.

Quadretic polynomial :

A polynomial of degree 2 is called quadretic polynomial. General form ofquadretic polynomial p(x)=ax2 + bx + c,Where a,b& c are real number and a ≠0.
Ex: p(x)= 4x2 + 5x+3 is an example of quadretic polynomial.

Cubic polynomial :

A polynomial of degree 3 is called cubic polynomial. General form of cubic polynomial p(x): ax3 + bx2 + cx + d, a ≠ 0, where a, b, and c are coefficients and d is the constant with all of them being real numbers.
Ex: p(x): x3 − 5x2 + 15x − 6 is an example of quadretic polynomial.

Zero of a polynomial:

Zeroes of a polynomial p(x) is a number k such that p(k) = 0.It is obtained by repalcaing x=k in polynomialP(X).Zero of a polynomial is also called the root of the polynomial.The maximum number of zeroes of a polynomial is equal to its degree.

In Liner polynomial p(x) only one value of x =k for which p(k)=0.
In Quadretic polynomial p(x) two values of x =k , l for which p(k)=0 & p(l)=0.
In Cubic polynomial p(x) three values of x =k,l,m for which p(k)=0, p(l)=0 & p(m)=0.

Value of polynomial:

If p(x) is a polynomialin xand if k is any real number then the value obtained by replacing x by k in p(x) is called value of p(x) at x=k. It is denoated p(k).


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