Understanding Polynomials and Algebraic Expressions
Dive into the world of algebraic expressions and polynomials, exploring concepts like constants, variables, coefficients, and degrees. Learn how to identify terms, understand polynomial types based on degrees, and solve related mathematical problems.
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CLASS IX , CHAPTER- POLYNOMIALS You have already studied about algebraic expressions in your previous classes. You know how to add, subtract, multiply or divide these. You have also studied about factorization using following identities: (a + b)2= a2+ 2ab + b2 (a - b)2= a2 2ab + b2 (a +b)(a b) = a2 b2 (a + b + c)2= a2+ b2+ c2+ 2ab + 2bc + 2ca (x + a)(x + b) = x2+ (a + b)x + ab Now, we will recall these concepts and extend them to expressions known as Polynomials.
Terms related to algebraic expressions and polynomials are CONSTANTS and VARIABLES: A variable is denoted by a symbol that can take any real value. We use the letters x, y, z etc. to denote variables. The values of the constants remain the same throughout a particular situation i.e. the values of the constants do not change in a given problem, but the value of the variable can keep changing. For example: Let s assume side of a square as a units. We know the perimeter of square is 4 x side = 4a units. As the side (i.e. a) varies, the perimeter( i.e. 4a) also varies. Hence, the side and perimeter of the square are variables whereas 4 is a constant in 4a.
ALGEBRAIC EXPRESSIONS: A combination of constants and variables, connected by some or all of the operations +, - , , is known as an algebraic expression. For example: x2 4x + 3 , x+2 etc. TERMS OF AN ALGEBRAIC EXPRESSION: The parts of an algebraic expression separated by +,- sign are called terms of an algebraic expression. For example: In algebraic expression x2 4x + 3, the terms are x2, -4x and 3. POLYNOMIAL: An algebraic expression in which the variables have only the non- negative integrals powers is called a polynomial. For example: 1) 5x3 + 4x2 6x 3 is a polynomial in one variable x 2) 3 + 2x2 6x2y + 5xy2is a polynomial in two variables x and y 3) 5 + 8x3/2 + 4x2 is an expression but not a polynomial since it contains a term with power of x equal 3/2 which is not an integer 4) 5x2 + 6x-1 - 3 is an expression but not a polynomial since it contains a term with power of x equal -1 which is not a non-negative integer
COEFFICIENTS: In the polynomial 5x3 - 4x2 - x 3, the coefficients of x3 , x2 and x are 5, -4 and -1 respectively whereas -3 is the constant term in this expression. 5 3x2 2 x + 3 expression has coefficients of x2 and x as 5 3 is the constant term. 3 and - 2 respectively whereas NOTE: Coefficients can be non-integral but the powers of the variables can only be non-negative integers DEGREE OF A POLYNOMIAL IN ONE VARIABLE: In case of a polynomial in one variable, the highest power of the variable is called the degree of the polynomial. For example: 1) 2x+ 3 is a polynomial in x of degree 1. 2) 5y3 4y2 - y 3 is a polynomial in y of degree 3. DEGREE OF A POLYNOMIAL IN TWO OR MORE VARIABLES: In case of polynomials in more than one variable, the sum of powers of the variables in each term is taken up and the highest sum so obtained is called the degree of the polynomial. For example: 1) 7x3 -5x2y2+ 3xy + 6y + 8 is a polynomial in x and y of degree 2+2 = 4 Now, watch the video: LINK: https://www.youtube.com/watch?v=MPcZ3nhZO-M
POLYNOMIALS OF VARIOUS DEGREES: TYPE OF POLYNOMIAL DEGREE EXAMPLE Linear 1 2x+ 3 is a linear polynomial in x Quadratic 2 2x2+ 3 is a quadratic polynomial in x Cubic 3 2x3 + 4x - 3 is a cubic polynomial in x Biquadratic 4 4x4 3x3 + 2x - 5 is a biquadratic polynomial in x
NUMBER OF TERMS IN A POLYNOMIALS: TYPE OF EXPRESSION NUMBER OF TERMS 1 EXAMPLE TERMS OF EXPRESSION Monomial i) 3x is a Monomial in x 3x Binomial 2 i) -2a + 3b is a Binomial in a and b i) ii) - 2a 3b Trinomial 3 i) 2p 3q + 7 is a Trinomial in p and q i) 2p ii) - 3q iii)7 LINK: https://www.youtube.com/watch?v=xmJjQ3KyTdw 1) CONSTANT POLYNOMIAL- A polynomial containing only one term that is a constant only is called a constant polynomial. For e.g. 3, 4/5, -2 etc. are constant polynomials. NOTE: The degree of a non-zero constant polynomial is zero. 2) ZERO POLYNOMIAL- A polynomial containing only one term namely 0 is called a zero polynomial. NOTE: The degree of a zero polynomial is not defined.
Now, watch the video : https://www.youtube.com/watch?v=D2t-RIZ0RBo
ZEROES OF A POLYNOMIAL: Let p(x) is a polynomial. If p(n) = 0, then we say that n is a zero of the polynomial p(x). Note that: To find the zeroes of a polynomial p(x) means solving the equation p(x) = 0 Now watch the link given below to understand it better LINK: https://www.youtube.com/watch?v=w0fVYd7c2Vs Example: Find a zero of the polynomial p(x) = x-3 (ii) q(x) = 5x + 3 Solution: (i) p(x) = x-3 p(x) = 0 gives x-3 =0 or x = 3 Therefore, 3 is a zero of given polynomial p(x) (ii) q(x) = 5x + 3 q(x) =0 gives 5x + 3 = 0 or 5x = -3 so, x = -3/5 Therefore, -3/5 is a zero of given polynomial q(x) NOTE THAT: (1) A non-zero constant polynomial has no zero. (2) Every real number is a zero of the zero polynomial. (i)
