Notes of Polynomials

Polynomials in One Variable

Polynomials

An expression of two or more than two algebraic terms that contain variable(s) that are raised to non-negative integral powers are called polynomials.

Types of Polynomials

Based on the number of terms a polynomial can be classified into monomial, binomial, trinomial,  etc.

  1. An algebraic expression having only one term is called a monomial. P(x)=x is a monomial.
  2. Polynomials having two terms are called binomials. P(x)=x2+2x has two terms, x2 and 2x.So, it is a binomial.
  3. Polynomials having three terms are called trinomials. P(x)=x4+3×2−4 has three terms, x4, 3×2 and −4.So, it is a trinomial.
  4. An algebraic expression of the form P(x)=c, where c is a constant is called constant polynomial.
  5. The constant polynomial 0 is called the zero polynomial.

Degree of a Polynomial

The degree of a polynomial is the highest degree of its individual terms with non-zero coefficients. The degree of a term is the sum of the exponents of the variables that appear in it. 

For a polynomial in one variable, the highest power of the variable in the polynomial is the degree of the polynomial. f(x)=x2−9×3+2×8−6 is a polynomial with degree 8 as the highest power to which x is raised is 8.

Note:

  • The degree of a non-zero constant polynomial is zero.
  • The degree of the zero polynomial is not defined.

A polynomial p(x) in one variable x is an algebraic expression in x of the form  p(x)=a xn n +a xn−1 n−1 +a xn−2 n−2 +……..+a x2 2 +ax a1 + 0, where a a a0 1 2, , ……an are constants i. 

x is a variable 

a ,a,a ……a0 1       2        n are respectively the coefficients of xi 

Each of a x a x a xn n, n−1 n−1, n−2 n−2,……..a x axa2 2, 1 , 0,with an ≠ 0,is called a term of a polynomial. 

The highest power of the variable in a polynomial is called the degree of the polynomial. 

  • A polynomial with one term is called a monomial
  • A polynomial with two terms is called a binomial
  • A polynomial with three terms is called a trinomial

A polynomial with degree zero is called a constant polynomial.

For example : 1, -3. The degree of non-zero constant polynomial is zero 

A polynomial of degree one is called a linear polynomial. It is of the form ax + b.

For example : x – 2, 4y + 89, 3x – z

A polynomial of degree two is called a quadratic polynomial. It is of the form ax2 + bx + c. where a, b,  c are real numbers and a≠ 0

For example : x2 − +2 5x  etc. 

A polynomial of degree three is called a cubic polynomial andhas the general form ax3 + bx2 + cx +d

For example : x3 + 2x2 − +2x 5 etc. 

A bi-quadratic polynomial p(x) is a polynomial of degree four which can be reduced to quadratic polynomial in the variable z = x2 by substitution. 

The zero polynomial is a polynomial in which the coefficients of all the terms of the variable are zero. Degree of zero polynomial is not defined. 

The value of a polynomial f(x) at x = p is obtained by substituting x = p in the given polynomial and is denoted by f(p). 

A real number ‘a’ is a zero/ root of a polynomial p(x) if p (a) = 0. 

The number of real zeroes of a polynomial is less than or equal to the degree of polynomial. 

Classification of Polynomials according to their Degree

Polynomials can be classified on the basis of their degree as follows:

  1. A polynomial of degree one is called a linear polynomial. P(x)=x−2 is a linear polynomial.
  2. A polynomial of degree two is called a quadratic polynomial. P(x)=x2−3x+4 is a quadratic polynomial.
  3. A polynomial of degree three a cubic polynomial. P(x)=x3+3x−2 is a cubic polynomial.

Representing equations on a graph

All polynomials can be represented on the graph to understand the nature of the polynomial, its zeroes etc.

For example, Geometrically zeros of a polynomial are the points where its graph cuts the x-axis.

Zeroes of a Polynomial

A zero of a polynomial P(x) is a number c such that P(c)=0.

The zero’s of the polynomial P(x)=x2−4 are 2 and (-2) since P(2)=(2)2−4=0 and P(−2)=(−2)2−4=0.

Note:

  • A non-zero constant polynomial has no zero.
  • Every real number is a zero of the zero polynomial.

Number of zeroes

In general, a polynomial of degree n has at most n zeros.

  • A linear polynomial has one zero.
  • A quadratic polynomial has at most two zeros.
  • A cubic polynomial has at most three zeros. 

Remainder Theorem

Long Division method to divide two polynomials

To divide one polynomial by another, follow the steps given below.

Step 1: arrange the terms of the dividend and the divisor in the decreasing order of their degrees. Suppose we want to divide (−x3+3×2+5−3x) by (−x2+x−1), we will arrange the terms of (−x3+3×2+5−3x) in decreasing order of their degrees as shown below.

Step 2: To obtain the first term of the quotient, divide the highest degree term of the dividend by the highest degree term of the divisor. Then carry out the division process. In our case, we will divide (−x3)(the highest degree term of the dividend) by (−x2)(the highest degree term of the divisor) to get the first term of the quotient. The first term of the quotient = −x3−x2 = x.

Division algorithm  

If p(x) and g(x) are the two polynomials such that degree of p(x) ≥ degree of g(x) and g(x) ≠ 0, then we can find polynomials q(x) and r(x) such that: 

p (x) = g(x) q(x) + r(x)   where, r(x) =0 or degree of r(x) < degree of g(x)

Step 3: The remainder of the previous division becomes the dividend for the next step. Repeat this process until the degree of the remainder is less than the degree of the divisor.

divide

Dividing one polynomial by another polynomial.

Remainder Theorem

When a polynomial f(x) of degree greater than or equal to one is divided by a linear polynomial x−a the remainder is equal to the value of f(a).

If f(a)=0 then x−a is a factor of the polynomial f(x).

Factor Theorem

If P(x) is a polynomial of degree greater than or equal to one and a is any real number then x−a is a factor of P(x) if P(a)=0.

Remainder theorem  

Let p(x) be any polynomial of degree greater than or equal to one and let a be any real number. If p(x) is divided by the linear polynomial (x – a), then remainder is p(a). 

If the polynomial p(x) is divided by (x+a), the remainder is given by the value of p (-a). ii. If p (x) is divided by ax + b = 0; a 0, the remainder is given by 

Screenshot 253

Factor theorem  

Let p(x) is a polynomial of degree n ≥ 1and a is any real number such that p(a) = 0, then (xa) is a factor of p(x). 

Converse of factor theorem  

Let p(x) is a polynomial of degree n ≥ 1and a is any real number. If (xa) is a factor of p(x), then    p(a) = 0. 

(x + a) is a factor of a polynomial p(x) iff p(-a) = 0. ii. (axb) is a factor of a polynomial p(x) iff p(b/a) = 0. iii. (ax + b) is a factor of a polynomial p(x) iff p(-b/a) = 0.  

(xa)(xb) is a factor of a polynomial p(x) iff p(a) = 0 and p(b) = 0. 

For applying factor theorem the divisor should be either a linear polynomial of the form (ax + b) or it should be reducible to a linear polynomial. 

If f(x) is a polynomial with integral coefficients and the leading coefficient is 1, then any integer root of f(x) is a factor of the constant term. 

Factorization of Polynomials

Factorisation of Quadratic Polynomials- Splitting the middle term

Factorisation of the polynomial ax2+bx+c by splitting the middle term is as follows:

Step 1: We split the middle term by finding two numbers such that their sum is equal to the coefficient of x and their product is equal to the product of the constant term and the coefficient of x2.

For example for the quadratic polynomial (x2+5x+6) the middle term can be split as, x2+2x+3x+6 

Here,2+3=5 and 2×3=6.

Step 2: Now, we factorise by pairing the terms and taking the common factors.

x2+2x+3x+6=x(x+2)+3(x+2)=(x+2)(x+3)

Thus, x+2 and x+3 are factors of x2+5x+6.

Factorisation of Quadratic Polynomials – Factor theorem

To factorise a quadratic polynomial f(x)=ax2+bx+c, find two numbers p and q such that f(p)=f(q)=0. Let us factorise the quadratic polynomial f(x)=x2−3x+2.

(i) f(2)=22−3(2)+2=4−6+2=0

Hence, x−2 is a factor of x2−3x+2.

(ii) f(3)=32−3×3+2=9−9+2=2≠0

Hence, x−3 is not a factor of x2−3x+2.

(iii) f(1)=12−3×1+2=0

Hence, x−1 is a factor of x2−3x+2.

So, x−1 and x−2 are the factors of the quadratic polynomial x2−3x+2.

x2−3x+2=(x−2)(x−1)

 A quadratic polynomial ax2 + bx+ c is factorised by splitting the middle term by writing b as ps + qr such that (ps) (qr) = ac

Then, ax2 + bx+ c = (px + q) (rx + s) 

An algebraic identity is an algebraic equation which is true for all values of the variables occurring in it. 

x3 + + −y3 z3         3xyz x y z x= + +( )( 2 + + − − −y2 z xy yz zx2

If x y z+ + = 0 then x3 + + =y3 z3 3xyz  Here, x, y and z are variables. 

Algebraic Identities

  • (a+b)2=a2+2ab+b2
  • (a-b)2=a2-2ab+b2
  • a2−b2=(a−b)(a+b)
  • (x+a)(x+b)=x2+(a+b)x+ab
  • (a+b+c)2=a2+b2+c2+2(ab+bc+ca)
  • (a+b)3=a3+b3+3ab(a+b)
  • (a−b)3=a3−b3−3ab(a−b)
  • a3+b3+c3−3abc=(a+b+c)(a2+b2+c2−ab−bc−ca)

Some useful quadratic identities

  • (x + y)2 = x2 + 2xy + y2
  • (x − y)2 = x2 – 2xy + y2 
  • (x y x y− )( + = −) x2 y2 
  • (x a x b x+ )( + = + +) 2 + (ab x ab)
  • (x + y + z)2 = x2 + y2 + z2+ 2xy + 2yz + 2zx
  • Here x, y, z are variables and a, b are constants.