POLYNOMIALS  

DEGREE OF POLYNOMIAL Degree of polynomial- The highest power of x in p(x) is called the degree of the polynomial p(x). EXAMPLE – 1) F(x) = 3x +½ is a polynomial in the variable x of degree 1. 2) g(y) = 2y²  ⅜ y +7 is a polynomial in the variable y of degree 2 .

TYPES OF POLYNOMIALS Types of polynomials are – 1] Constant polynomial 2] Linear polynomial 3] Quadratic polynomial 4] Cubic polynomial

CONSTANT POLYNOMIAL CONSTANT POLYNOMIAL – A polynomial of degree zero is called a constant polynomial. EXAMPLE - p(x) = 7 etc .

LINEAR POLYNOMIAL  

QUADRATIC POLYNOMIAL QUADRATIC POLYNOMIAL – A polynomial of degree 2 is called quadratic polynomial . EXAMPLE – 2x²  3x  ⅔ , y²  2 etc . More generally , any quadratic polynomial in x with real coefficient is of the form ax² + bx + c , where a, b ,c, are real numbers and a  0

CUBIC POLYNOMIALS CUBIC POLYNOMIAL – A polynomial of degree 3 is called a cubic polynomial . EXAMPLE = 2  x³ , x³, etc . The most general form of a cubic polynomial with coefficients as real numbers is ax³  bx²  cx  d , a ,b ,c ,d are reals and a  0

VALUE OF POLYNOMIAL If p(x) is a polynomial in x, and if k is any real constant, then the real number obtained by replacing x by k in p(x), is called the value of p(x) at k, and is denoted by p(k) . For example , consider the polynomial p(x) = x² 3x 4 . Then, putting x= 2 in the polynomial , we get p(2) = 2²  3  2  4 =  4 . The value  6 obtained by replacing x by 2 in x²  3x  4 at x = 2 . Similarly , p(0) is the value of p(x) at x = 0 , which is  4 .

ZERO OF A POLYNOMIAL  

HOW TO FIND THE ZERO OF A LINEAR POLYNOMIAL In general, if k is a zero of p(x) = ax  b, then p(k) = ak  b = 0, k =  b  a . So, the zero of a linear polynomial ax  b is  b  a =  ( constant term )  coefficient of x . Thus, the zero of a linear polynomial is related to its coefficients .

GEOMETRICAL MEANING OF THE ZEROES OF A POLYNOMIAL We know that a real number k is a zero of the polynomial p(x) if p(K) = 0 . But to understand the importance of finding the zeroes of a polynomial, first we shall see the geometrical meaning of – 1) Linear polynomial . 2) Quadratic polynomial 3) Cubic polynomial

GEOMETRICAL MEANING OF LINEAR POLYNOMIAL For a linear polynomial ax  b , a  0, the graph of y = ax b is a straight line . Which intersect the x axis and which intersect the x axis exactly at one point ( b  a , 0 ) . Therefore the linear polynomial ax  b , a  0 has exactly one zero .

QUADRATIC POLYNOMIAL For any quadratic polynomial ax²  bx c, a  0, the graph of the corresponding equation y = ax²  bx  c has one of the two shapes either open upwards or open downward depending on whether a0 or a0 .these curves are called parabolas .

GEOMETRICAL MEANING OF CUBIC POLYNOMIAL The zeroes of a cubic polynomial p(x) are the x coordinates of the points where the graph of y = p(x) intersect the x – axis . Also , there are at most 3 zeroes for the cubic polynomials . In fact, any polynomial of degree 3 can have at most three zeroes .

RELATIONSHIP BETWEEN ZEROES OF A POLYNOMIAL For a quadratic polynomial – In general, if  and  are the zeroes of a quadratic polynomial p(x) = ax²  bx  c , a  0 , then we know that x   and x  are the factors of p(x) . Therefore , ax²  bx  c = k ( x  ) ( x   ) , Where k is a constant = k[x²  (  )x ] = kx²  k(    ) x  k  Comparing the coefficients of x² , x and constant term on both the sides . Therefore , sum of zeroes =  b  a =  (coefficients of x)  coefficient of x² Product of zeroes = c  a = constant term  coefficient of x²

RELATIONSHIP BETWEEN ZERO AND COEFFICIENT OF A CUBIC POLYNOMIAL In general, if  ,  , Y are the zeroes of a cubic polynomial ax³  bx²  cx  d , then Y =  ba =  ( Coefficient of x² )  coefficient of x³  Y Y =c  a = coefficient of x  coefficient of x³ Y =  d  a =  constant term  coefficient of x³

DIVISION ALGORITHM FOR POLYNOMIALS If p(x) and g(x) are any two polynomials with g(x)  0, then we can find polynomials q(x) and r(x) such that – p(x) = q(x)  g(x)  r(x) Where r(x) = 0 or degree of r(x)  degree of g(x) . This result is taken as division algorithm for polynomials .

Steps for dividing one polynomial by another We first arrange the terms of the dividend and the divisor in the decreasing order of their degrees. To obtain the first term of the quotient, divide the highest degree term of the dividend by the highest degree term of the divisor. Then carry out the division process. If the degree of the new dividend is less than the degree of the divisor, then stop. Else, to obtain the second term of the quotient, divide the highest degree term of the new dividend by the highest degree term of the divisor. Again carry out the division process, if possible.

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