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- 1. Made By : Sejal Agarwal School : Ryan International School Class : 9th Subject : Maths Submitted To : Ms Madhu Kanta
- 2. Algebraic Expression : The combination of constants and variables are called algebraic expressions. E.g.- (a) 2x3 –4x2 +6x–3 is a polynomial in one variable x. (b) 8p7+4p2+11p3-9p is a polynomial in one variable p. (c) 4+7x4/5 +9x5 is an expression but not a polynomial since it contains a term x4/5 , where 4/5 is not a non-negative integer.
- 3. Polynomials : An algebraic expression in which the variable involved have only non –negative integral powers is called a polynomial. Eg. : 3x + 4y Constants : A symbol having a fixed numerical value is called a constant. Eg. : In polynomial 3x + 4y ,3 and 4 are the constants. Variables : A symbol which may be assigned different numerical values is called as a variables. Eg. : In polynomial 3x + 4y , x and y are the variables.
- 4. Degree : The highest power of a variable in the polynomial is called degree of that polynomial. Eg. : 5x2 + 3 , here the degree is 2. Constant polynomial : A polynomial containing one term only , consisting of a constant is called a constant polynomial. The degree of a nonzero constant polynomial is zero. Eg. : 3 , -5 , 7/8 , etc. , are all constant polynomials. Zero polynomial : A polynomial consisting one term only , namely zero only , is called a zero polynomial. The degree of a zero polynomial is not defined.
- 5. Monomial : Algebric expression that consists only one term is called monomial. Binomial : Algebric expression that consists two terms is called binomial. Trinomial : Algebric expression that consists three terms is called trinomial. Polynomial : Algebric expression that consists many terms is called polynomial.
- 6. Types of polynomial on the basis of degree are : Linear polynomial: A polynomial of degree 1 is called a linear polynomial. Quadratic polynomial: A polynomial of degree 2 is called a quadratic polynomial. Cubic polynomial : A polynomial of degree 3 is called a cubic polynomial. Biquadratic polynomial : A polynomial of degree 4 is called a biquadratic polynomial.
- 7. Polynomials Degree Classify by degree Classify by no. of terms. 5 0 Constant Monomial 2x - 4 1 Linear Binomial 3x2 + x 2 Quadratic Binomial x3 - 4x2 + 1 3 Cubic Trinomial
- 8. Phase 1Phase 1 Phase 2Phase 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?
- 9. 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).
- 10. Let f(x) be a polynomial of degree n > 1 and let a be any 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).
- 11. 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)
- 12. 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)
- 13. 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.

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