Chapter 2 Polynomials Notes Class 9

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POLYNOMIALS 

1.Expression: An expression which is the combination of constants and variables and are connected by  some or all the operations addition, subtraction, multiplication and division is known as an  algebraic expression. 

Example: 7 + 9x – 2x2 +56xy 

2. Constant: Which has fixed numerical value. 

Example: 7, -4, 3/4, etc. 

3. Variable: A symbol which has no fixed numerical value is known as a variable. Example: 2x, 5x2 

4. Terms: These are the parts of an algebraic expression which are separated by operations, like  addition or subtraction are known as terms. 

Example: In the expression 5x3 + 9x2 + 7x – 3, terms are 5x3, 9x2, 7x and -3 

5. Polynomial: An algebraic expression of which variables have non-negative integral powers is  called a polynomial. 

Example: 

(a) 5x2 + 7x + 3 

(b) 9y3 – 7y2 + 3y + 7  

6. Coefficient: A coefficient is the numerical value in a term. 

Note: If a term has no coefficient, the coefficient is an unwritten 1. 

Example: 5x3 – 7x2 – x + 3 

7. Degree of a polynomial (in one variable): The highest power of the variable is called the  degree of the polynomial. 

Example: 5x + 4 is a polynomial in x of degree 1. 

8. Degree of a polynomial in two or more variables: The highest sum of powers of variables is  called the degree of the polynomial. 

Example: 7x3 + 2x2y2 – 3xy + 8 

9.Degree of polynomial = 4 (Sum of the powers of variables x and y ) 

10. Types of Polynomial 

(i) Linear polynomial: A polynomial of degree one is called a linear polynomial. Example: 2x + 3 is a linear polynomial in x. 

(ii) Quadratic polynomial: A polynomial of degree 2 is called a quadratic polynomial. Example: 5x2 – 7x + 4 is a quadratic polynomial. 

(iii) Cubic polynomial: A polynomial of degree 3 is called a cubic polynomial. Example: 3x3 + 7x2 – 4x + 9 is a cubic polynomial. 

(iv) Biquadratic polynomial: A polynomial of degree 4 is called a biquadratic polynomial. Example: 7x4 – 2x3 + 4x + 9 is a biquadratic polynomial.

11. Number of Terms in a Polynomial 

Categories of the polynomial according to their terms: 

(i) Monomial A polynomial which has only one non-zero term is called a monomial. Example: 7, 4x, 45xy, 7x2y3z5, all are monomials. 

(ii) Binomial: A polynomial which has only two non-zero terms is called binomial. Example: 2x + 7, 9x2 + 3, 3x2yz + 4x3y3z2, all are binomials. 

(iii) Trinomial: A polynomial which has only three non-zero terms is called a trinomial. Example: 5x2 + lx + 9, 5xy + 7xy2 + 3x3yz, all are trinomials. 

(iv) Constant polynomial: A polynomial which has only one term and that is a constant is called  a constant polynomial. 

Example: −34, 7, 5 all are constant polynomials.  

Note: The degree of constant non-zero polynomial is zero. 

(v) Zero polynomial. A polynomial which has only one term i.e., 0 is called a zero polynomial. Note: Degree of a zero polynomial is not defined. 

12. Value of a Polynomial 

Value of a polynomial is obtained, when variable of a given polynomial is interchanged or  replaced by a ; constant. Let p(x) is a polynomial then value of polynomial at x = a is p(a). Zero or root of a polynomial: A zero or root of a polynomial is the value of that variable for  which value of polynomial p(x) becomes zero i.e., p(x) = 0. 

Let p(x) be the polynomial and x – a. 

If p(a) = 0 then real value a is called zero of a polynomial. 

13. Remainder Theorem 

Let p(x) be a polynomial of degree ≥ 1 and a be any real number. If p(x) is divided by the linear  polynomial x-a, then the remainder is p(a). 

Proof: Let p(x) be any polynomial of degree greater than or equal to 1. When p(x) is divided by  x – a, the quotient is q(x) and remainder is r(x). 

i.e.,p(x) = (x-a) q(x) + r(x) 

Since degree of x – a is 1 and the degree of r(x) is less than the degree of x – a so the degree of  r(x) = 0. 

It: means r(x) is a constant, say r. 

Therefore, for every value of x, r(x) = r 

then p(x) = (x-a) q(x) + r 

When x = a, then p(a) = (a – a) q(x) + r ⇒ p(a) = r 

14. Factor Theorem 

If p(x) is a polynomial of degree greater than or equal to 1 and a be any real number, then 

• x – a is a factor of p(x) i.e., p(x) = (x-a) q(x) which shows x – a is a factor of p(x) • Since x – a is a factor of p(x) 

p(x) = (x-a).g(x) for same polynomial g(x). In this case, p(a) =(a-a) g (a) = 0

15. Factorisation of the Polynomial ax2 + bx + c by Splitting the Middle Term Let p(x) = ax2 + bx + c and factor of polynomial p(x) = (px + q) and (rx + s)

then ax2 + bx + c = (px + q) (rx + s) = prx2 + (ps +qr)x+ qs 

Comparing the coefficient of x2 on both sides 

a = pr …………. (1) 

Comparing the coefficient of x 

b =ps + qr …………. (2) 

and comparing the constant terms 

c = qs ……………..(3) 

which shows that b is the sum of two numbers ps + qr. 

Product of two numbers ps x qr =pr x qs = ac 

So for factors ax2 + bx + c, we should write b as sum of two numbers whose product is ac. Example: Factorise 6x2 + 17x + 5 

Here, b = p + q = 17 

and ac = 6 x 5 = 30 (= pq) 

then we get factors of 30, 1 x 30, 2 x 15, 3 x 10, 5 x 6, 

Among above factors of 30, the sum of 2 and 15 is 17 

i.e.,p + q = 2 + 15 = 17 

∴ 6x2 + 17x + 5 = 6x2 + (2 + 15)x + 5 = 6x2 + 2x + 15x + 5 

= 2x(3x + 1) + 5(3x + 1) = (3x + 1) (2x + 5) 

16. Algebraic Identities  

1. (a + b)2 = a2 + 2ab + b2 

2. (a – b)2 = a2 – 2ab + b2 

3. (a + b) (a – b) = a2-b2 

4. (x + a) (x + b) = x2 + (a + b) x + ab 

5. (x + a) (x – b) = x2 + (a – b) x – ab 

6. (x – a) (x + b) = x2 + (b – a) x – ab 

7. (x – a) (x – b) = x2 – (a + b) x + ab 

8. (a + b)3 = a3 + b3 + 3ab (a + b) 

9. (a – b)3 = a3 – b3 – 3ab (a – b) 

10. (x + y + z)2 = x2 + y2 + z2 + 2xy +2yz + 2xz 

11. (x + y – z)2 = x2 + y2 + z2 + 2xy – 2yz – 2xz 

12. (x – y + z)2 = x2 + y2 + z2 – 2xy – 2yz + 2xz 

13. (x – y – z)2 = x2 + y2 + z2 – 2xy + 2yz – 2xz 

14. x3 + y3 + z3 – 3xyz = (x + y + z) (x2 + y2 + z2 – xy – yz -xz) 15. x2 + y2 = 12[(x + y)2 + (x – y)2] 

16. (x + a) (x + b) (x + c) = x3 + (a + b + c)x2 + (ab + bc + ca)x + abc 17. x3 + y3 = (x + y) (x2 – xy + y2) 

18. x3 – y3 = (x – y) (x2 + xy + y2) 

19. x2 + y2 + z2 – xy – yz – zx = 12[(x – y)2 + (y – z)2 + (z – x)2]