Topics Covered• Polynomials.• Exponents And Terms.• Degree Of A Polynomial In One Variable.• Degree Of A Polynomial In Two Variables.• Remainder Theorem.• Factor Theorem.• Algebric Identities.
Polynomials• 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.
Degree of a Polynomial in one variable:-Degree of a Polynomial in one variable. Whatis degree of the following binomial? Theanswer is 2. 5x2 + 3 is a polynomial in x ofdegree 2. In case of a polynomial in onevariable, the highest power of the variable iscalled the degree of polynomial .
Degree of a Polynomial in two variables. • What is degree of the following polynomial? 5 x y − 7 x + 3 xy + 9 y + 4 2 3 3• Theanswer is five because if we add 2 and 3 , the answer is fivewhich is the highest power in the whole polynomial.E.g.- 3 x y − 5 x + 8 xy + 2 y + 9 3 4 2 is a polynomial in x and y of degree 7.In case of polynomials on more than one variable, the sum ofpowers of the variables in each term is taken up and the highestsum so obtained is called the degree of polynomial.
Polynomials in one variable The degree of a polynomial in one variable is the largest exponent of that variable.2 A constant has no variable. It is a 0 degree polynomial.4x +1 This is a 1st degree polynomial. 1st degree polynomials are linear.5 x + 2 x − 14 2 This is a 2nd degree polynomial. 2nd degree polynomials are quadratic.3 x − 18 This is a 3rd are cubic. 3 polynomials degree polynomial. 3rd degree
ExamplesPolynomials Degree Classify by Classify by no. degree of terms. Text 5 0 Constant Monomial Txt 2x - 4 1 Linear Binomial Text 3x2 + x 2 Quadratic Binomial Text Textx - 4x + 1 3 2 3 Cubic Trinomial
Remainder TheoremLet 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). TEXT TEXT TEXT TEXTPROOF 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 TheoremQ.) Find the remainder when the polynomial f(x) = x4 + 2x3 – 3x2 + x – 1 is divided by (x-2).A.) x-2 = 0 x=2By 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 TheoremLet f(x) be a polynomial of degree n > 1 and let a beany real number.(i) 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).
Algebraic Identities Some common identities used to factorize polynomials(a+b)2=a2+b2+2ab (a-b)2=a2+b2-2ab a2-b2=(a+b)(a-b) (x+a)(x+b)=x2+(a+b)x+ab
Algebraic Identities Advanced identities used to factorize polynomials (x-y)3=x3-y3- 3xy(x-y)x3+y3=(x+y) * (x2+y2-xy) x3-y3=(x+y) * (x2+y2+xy) (x+y)3=x3+y3+ (x+y+z)2=x2+y2+z2 3xy(x+y) +2xy+2yz+2zx
Q/A on PolynomialsQ.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).
Points to Remember• A real number ‘a’ is a zero of a polynomial p(x) if p(a)=0. In this case, a is also called a root of the equation p(x)=0.• Every linear polynomial in one variable has a unique zero, a non-zero constant polynomial has no zero, and every real number is a zero of the zero polynomial.