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## POLYNOMIALS

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**POLYNOMIAL IN ONE VARIABLE**• For real variable x, an expression of the type P (x) = a0 + a1x + a2 x2 + ----- + an x n with real co-efficient a0, a1, a2, an and n is positive integer is the real polynomial in one variable.**POLYNOMIAL EQUATIONS**• If P (x) is a real polynomial then P (x) = 0 is a polynomial equation. • 2x + 3 is a polynomial and 2x + 3 = 0 is polynomial equation. • 2x -1 + 3 is not a polynomial but 2x -1 + 3 = 0 is still a polynomial equation.**ZERO OF POLYNOMIAL**• The value of variable x = a is called the zero of polynomial p (x) if P (a) = 0.**ZERO OF POLYNOMIAL**• The zero of an polynomial are the x co-ordinates (Abscissa) of the points where curve y = f (x) crosses the x-axis. e.g. for x + 1 consider y = x + 1 X = -1 is the zero of the polynomial x + 1. 1 X o -1 Y/**ZERO OF POLYNOMIAL**• For polynomial x2 -3x + 2 • Consider y = x2 -3x + 2 The zeros of the polynomial are 1 and 2. X o 1 2 Y/**TYPES OF POLYNOMIALS**• Polynomials are named based on two criterions 1. number of terms the polynomial has. 2. the highest exponent of the variable present in the polynomial.**GRAPHICAL REPRESENTATION OF LINEAR POLYNOMIAL**• For linear polynomial a x + b , a ≠ 0, consider y = a x + b. • Linear polynomial represent a straight line intersecting x-axis at point (-b/a, 0) and y-axis at (0,b) Y For - 2x + 4, curve is a straight line interesting x-axis at (2,0) and y-axis at (0,4) 4 X O 2**QuadraticPolynomial**Polynomial in one variable is called quadratic polynomial if a3, a 4 ,------ = 0 , a 2≠ 0 and a 0, a 1 may or may not be zero. In general, a quadratic polynomial is denoted as a x2 + b x + c , with a ≠ 0 X2 –5x +4**GRAPHICAL REPRESENTATION OF QUADRATIC POLYNOMIAL**• For quadratic polynomial a x2 + b x + c , a ≠ 0, consider y = a x2 + b x + c.**GRAPHICAL REPRESENTATION OF QUADRATIC POLYNOMIAL**• For quadratic polynomial a x2 + b x + c , a ≠ 0, consider y = a x2 + b x + c.**GRAPHICAL REPRESENTATION OF QUADRATIC POLYNOMIAL**• For quadratic polynomial a x2 + b x + c , a ≠ 0, consider y = a x2 + b x + c.**GRAPHICAL REPRESENTATION OF QUADRATIC POLYNOMIAL**• For quadratic polynomial a x2 + b x + c , a ≠ 0, consider y = a x2 + b x + c.**GRAPHICAL REPRESENTATION OF QUADRATIC POLYNOMIAL**• For quadratic polynomial a x2 + b x + c , a ≠ 0, consider y = a x2 + b x + c. Put b2 – 4ac = D**GRAPHICAL REPRESENTATION OF QUADRATIC POLYNOMIAL**• For quadratic polynomial a x2 + b x + c , a ≠ 0, consider y = a x2 + b x + c. It is a parabola with vertex at (-b/2a,-D/4a)**GRAPHICAL REPRESENTATION OF QUADRATIC POLYNOMIAL**• For quadratic polynomial a x2 + b x + c , a ≠ 0, consider y = a x2 + b x + c. If a > 0 No Zeros**GRAPHICAL REPRESENTATION OF QUADRATIC POLYNOMIAL**• For quadratic polynomial a x2 + b x + c , a ≠ 0, consider y = a x2 + b x + c. If a < 0 No Zeros**SIGN OF QUADRATIC POLYNOMIAL**FOR THE EXPRESSION F(X) = a x 2 + b x + c, a ≠ 0 CASE I For D = b 2 – 4 ac < 0 If a > 0 then f (x) > 0 for all real values of x. If a < 0 then f (x) < 0 for all real values of x. a < 0, f (x) < 0 a > 0, f (x) > 0**SIGN OF QUADRATIC POLYNOMIAL**FOR THE EXPRESSION F(X) = a x 2 + b x + c, a ≠ 0 CASE II For D = b 2 – 4 ac = 0 If a > 0 then f (x) ≥ 0 for all real values of x. If a < 0 then f (x) ≤ 0 for all real values of x. a < 0, f (x) ≤ 0 a > 0, f (x) ≥ 0**SIGN OF QUADRATIC POLYNOMIAL**FOR THE EXPRESSION F(X) = a x 2 + b x + c, a ≠ 0 CASE III For D = b 2 – 4 ac > 0 If a > 0 then f (x) ≥ 0 for all real values of x. If a < 0 then f (x) ≤ 0 for all real values of x. a < 0, f (x) ≤ 0 a > 0, f (x) ≥ 0**SIGN OF QUADRATIC POLYNOMIAL**FOR THE EXPRESSION F(X) = a x 2 + b x + c, a ≠ 0 CASE III For D = b 2 – 4 ac > 0 If a > 0 then Y > 0 for x < α , x > β f (x) = = 0 for x = α , β < 0 for α < x > β X α β**SIGN OF QUADRATIC POLYNOMIAL**FOR THE EXPRESSION F(X) = a x 2 + b x + c, a ≠ 0 CASE III For D = b 2 – 4 ac > 0 If a < 0 then f (x) Y < 0 for x < α , x > β f (x) = = 0 for x = α , β > 0 for α < x > β X α β**CONCLUSION**• Polynomial f (x) = a x2 + b x + c has the same sign as that of “a” except when zeros of quadratic polynomial are real and distinct and x lies between them.**Descartes' Rule Of Signs**• The maximum number of positive real zeros of a polynomial f (x) is the number of changes of sign from positive to negative and vice versa in f (x). e.g. in case of x3 – 2 x2 – x + 2 [ = ( x - 1) (x - 2) ( x +1)] + - - + Here zeros are 1, 2 & -1 1st 2 nd Positive zeros**The maximum number of negative real zeros of a**polynomial f (x) is the number of changes of sign from positive to negative and vice versa in f (- x). e.g. in case of x3 – 2 x2 – x + 2 [ = ( x - 1) (x - 2) ( x +1)] f(- x) = - x3 – 2 x2 + x + 2 Here zeros are 1, 2 & -1 - + + + 1st Negative zeros**POSITIVE OR NEGATIVE ZEROS**• X3 +2X2-9X-18 Find positive and negative zeros. • If the remainder on division of x3+2x2+kx+3 by x-3 is 21,find the quotient and the value of k.Hence ,find the zeros of the cubic polynomial x3+2x2+kx-18 Ans (3,-2,-3)**QUESTIONS**• Find k so that x2+2x+k is a factor of 2x4+x3-14x2+5x+6.Also find all the zeros of the polynomials.(-3: -3,1,2,-1/2; 1,-3) • Given that the zeros of the cubic polynomial x3-6x2+3x+10 are of the form a,a+b,a+2b for some real numbers a and b,find the values of a and b as well as the zeros of the given polynomials.(-1,2,5)