POLYNOMIALS: POLYNOMIALS INTRODUCTION: We have studied algebraic expressions, their addition, subtraction, multiplication and division in earlier classes. We also have studied how to factories some algebraic expressions. We may recall the algebraic identities: (x + y )2=x2+2xy+y2 (x - y)=x2 -2xy+y2 and x2-y2=(x+y)(x-y) And their use in factorization. In this Power Point Presentation with a particular type of algebraic expression, called Polynomial, and the terminology related to it. We shall also see the Remainder Theorem and Factor theorem and their use in the factorization of polynomials. In addition to the above , we shall see some more algebraic identities and their use in factorization and in evaluating some given expressions. INTRODUCTION CONTENTS: CONTENTS DEGREE OF A PLYNOMIALS IN ONE VARIABLE DEGREE OF A POLYNOMIAL IN TWO VARIABLE POLYNOMIALS IN ONE VARIABLES POLYNOMIAL IN TWO VARIABLE A POLYNOMIAL HAVE Constants Variables Exponents Coefficients Degree of Polynomials Types of Polynomials Linear Polynomials Quadratic Polynomial Cubic Polynomials BIQUADRATIC POLYNOMIALS NON-ZERO POLYNOMIALS ZERO POLYNOMIAL MONOMIAL, BINOMIAL AND TRINOMIAL DEGREE OF A PLYNOMIALS IN ONE VARIABLE : DEGREE OF A PLYNOMIALS IN ONE VARIABLE Degree of a Polynomial in one variable. : Degree of a Polynomial in one variable. What is degree of the following binomial? The answer is 2. 5x2 + 3 is a polynomial in x of degree 2. In case of a polynomial in one variable, the highest power of the variable is called the degree of polynomial DEGREE OF A POLYNOMIAL IN TWO VARIABLE: DEGREE OF A POLYNOMIAL IN TWO VARIABLE Degree of a Polynomial in two variables. : Degree of a Polynomial in two variables. What is degree of the following polynomial? In case of polynomials on 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 polynomial. The answer is five because if we add 2 and 3 , the answer is five which is the highest power in the whole polynomial. E.g.- is a polynomial in x and y of degree 7 POLYNOMIALS IN ONE VARIABLES : POLYNOMIALS IN ONE VARIABLES Polynomials in one variable A polynomial is a monomial or a sum of monomials. Each monomial in a polynomial is a term of the polynomial. The number factor of a term is called the coefficient. The coefficient of the first term in a polynomial is the lead coefficient. A polynomial with two terms is called a binomial. A polynomial with three term is called a trinomial POLYNOMIAL IN TWO VARIABLE: POLYNOMIAL IN TWO VARIABLE Polynomials in one variable The degree of a polynomial in one variable is the largest exponent of that variable. A constant has no variable. It is a 0 degree polynomial. This is a 1st degree polynomial. 1st degree polynomials are linear. This is a 2nd degree polynomial. 2nd degree polynomials are quadratic. This is a 3rd degree polynomial. 3rd degree polynomials are cubic PowerPoint Presentation: Standard Form : Standard Form Phase 1 Phase 2 To rewrite a polynomial in standard form, rearrange the terms of the polynomial starting with the largest degree term and ending with the lowest degree term. The leading coefficient, the coefficient of the first term in a polynomial written in standard form, should be positive. How to convert a polynomial into standard form? Remainder Theorem : Remainder Theorem :- Let f(x) be a polynomial of degree n > 1 and let a be any real number. When f(x) is divided by (x-a) , then the remainder is f(a). PROOF Suppose when f(x) is divided by (x-a), the quotient is g(x) and the remainder is r(x). Then, degree r(x) < degree (x-a) degree r(x) < 1 [ therefore, degree (x-a)=1] degree r(x) = 0 r(x) is constant, equal to r (say) Thus, when f(x) is divided by (x-a), then the quotient is g9x) and the remainder is r. Therefore, f(x) = (x-a)*g(x) + r ( i ) Putting x=a in ( i ), we get r = f(a) Thus, when f(x) is divided by (x-a), then the remainder is f(a). Questions on Remainder Theorem : Questions on Remainder Theorem Q.) Find the remainder when the polynomial f(x) = x4 + 2x3 – 3x2 + x – 1 is divided by (x-2). A.) x-2 = 0 x=2 By remainder theorem, we know that when f(x) is divided by (x-2), the remainder is x(2). Now, f(2) = (24 + 2*23 – 3*22 + 2-1) = (16 + 16 – 12 + 2 – 1) = 21. Hence, the required remainder is 21. Factor Theorem : Factor Theorem Let f(x) be a polynomial of degree n > 1 and let a be any real number. If f(a) = 0 then (x-a) is a factor of f(x). PROOF let f(a) = 0 On dividing f(x) by 9x-a), let g(x) be the quotient. Also, by remainder theorem, when f(x) is divided by (x-a), then the remainder is f(a). therefore f(x) = (x-a)*g(x) + f(a) f(x) = (x-a)*g(x) [therefore f(a)=0(given] (x-a) is a factor of f(x). POLYNOMIALS PowerPoint Presentation: Algebraic Identities : Algebraic Identities Some common identities used to factorize polynomials (x+a)(x+b)=x2+(a+b)x+ab (a+b)2=a2+b2+2ab (a-b)2=a2+b2-2ab a2-b2=(a+b)(a-b) Algebraic Identities : Algebraic Identities Advanced identities used to factorize polynomials (x+y+z)2=x2+y2+z2+2xy+2yz+2zx (x-y)3=x3-y3-3xy(x-y) (x+y)3=x3+y3+3xy(x+y) x3+y3=(x+y) * (x2+y2-xy) x3-y3=(x+y) * (x2+y2+xy) Q/A ON POLYNOMIALS Q/A on Polynomials Q.1) Show that (x-3) is a factor of polynomial f(x)=x3+x2-17x+15. A.1) By factor theorem, (x-3) will be a factor of f(x) if f(3)=0. Now, f(x)=x3+x2-17x+15 f(3)=(33+32-17*3+15)=(27+9-51+15)=0 (x-3) is a factor of f(x). Hence, (x-3) is a factor of the given polynomial f(x). Q/A on Polynomials : Q/A on Polynomials Q.1) Factorize: ( i ) 9x2 – 16y2 (ii)x3-x A.1)( i ) (9x2 – 16y2) = (3x)2 – (4y)2 = (3x + 4y)(3x – 4y) therefore, (9x2-16y2) = (3x + 4y)(3x – 4y) (ii) (x3-x) = x(x2-1) = x(x+1)(x-1) therefore, (x3-x) = x(x + 1)(x-1) POLYNOMIALS Constants: Constants In mathematics, a constant is a non-varying value, i.e. completely fixed or fixed in the context of use. The term usually occurs in opposition to variable (i.e. variable quantity), which is a symbol that stands for a value that may vary. For e.g. 2x 2 +11y-22=0 , here -22 is a constant. Variables: Variables In mathematics, a variable is a value that may change within the scope of a given problem or set of operations. In contrast, a constant is a value that remains unchanged, though often unknown or undetermined. For e.g. 10x 2 +5y=2 , here x and y are variable. Exponents: Exponents Exponents are sometimes referred to as powers and means the number of times the 'base' is being multiplied. In the study of algebra, exponents are used frequently. For e.g.- Coefficients: Coefficients For other uses of this word, see coefficient (disambiguation).In mathematics, a coefficient is a multiplicative factor in some term of an expression (or of a series); it is usually a number, but in any case does not involve any variables of the expression. For e.g.- 7x 2 − 3xy + 15 + y Here 7, -3, 1 are the coefficients of x 2 , xy and y respectively. Degree of Polynomials : Degree of Polynomials The degree of a polynomial is the highest degree for a term. The degree of a term is the sum of the powers of each variable in the term. The word degree has for some decades been favoured in standard textbooks. In some older books, the word order is used. For e.g.- The polynomial 3 − 5x + 2x 5 − 7x 9 has degree 9 . TYPES OF POLYNOMIAL: TYPES OF POLYNOMIAL DEGREE NAME EXAMPLE 1 Linear X+1 2 Quadratic X 2 +1 3 Cubic X 3 +1 4 Quartic(Biquadratic) X 4 +1 Linear Polynomials: Linear Polynomials In a different usage to the above, a polynomial of degree 1 is said to be linear, because the graph of a function of that form is a line. For e.g.- 2x+1 11y +3 Quadratic Polynomials: Quadratic Polynomials In mathematics, a quadratic polynomial or quadratic is a polynomial of degree two, also called second-order polynomial. That means the exponents of the polynomial's variables are no larger than 2 . For e.g.- x 2 − 4 x + 7 is a quadratic polynomial, while x 3 − 4 x + 7 is not. Cubic Polynomials: Cubic Polynomials Cubic polynomial is a polynomial of having degree of polynomial no more than 3 or highest degree in the polynomial should be 3 and should not be more or less than 3 . For e.g.- x 3 + 11x = 9x 2 + 55 x 3 + x 2 +10x = 20 Biquadratic Polynomials : Biquadratic Polynomials Biquadratic polynomial is a polynomial of having degree of polynomial is no more than 4 or highest degree in the polynomial is not more or less than 4 . For e.g.- 4x 4 + 5x 3 – x 2 + x - 1 9y 4 + 56x 3 – 6x 2 + 9x + 2 Types Of Polynomial: Number of non- zero terms Name Example 0 Zero Polynomial 0 1 Monomial X 2 2 Binomial X 2 +1 3 Trinomial X 3 +1 Types Of Polynomial Polynomial can be classified by number of non-zero term Zero Polynomials: Zero Polynomials The constant polynomial whose coefficients are all equal to 0 . The corresponding polynomial function is the constant function with value 0 , also called the zero map . The degree of the zero polynomial is undefined, but many authors conventionally set it equal to -1. Monomial, Binomial & Trinomial: Monomial, Binomial & Trinomial Monomial :- A polynomial with one term . E.g. - 5x 3 , 8 , and 4xy . Binomial :- A polynomial with two terms which are not like terms. E.g. - 2x – 3, 3x 5 +8x 4 , and 2ab – 6a 2 b 5 . Trinomial :- A polynomial with three terms which are not like terms. E.g. - x 2 + 2x - 3, 3x 5 - 8x 4 + x 3 , and a 2 b + 13x + c .