Page 1 POLYNOMIALS GRAPHS OF POLYNOMIALS 1) Draw a graph of the polynomial f x x Solution: Let y = 3 2 x+ now we get different values of y corresponding to the different values of x. x 0 1 2 y 2 5 8 The points A(0,2), B(1,5) and C(2,8) are plotted on the graph paper on a suitable scale. The points are joint together to get the graph of the given polynomial. Since the polynomial is linear we get the graph as a straight line. RELATIONSHIP BETWEEN ZEROES AND COEFFICIENTS OF A POLYNOMIAL 2) Find the zeroes of the quadratic polynomial between the zeroes and the coefficients. Solution: The given quadratic polynomial can be Thus the value of 2 12 32 x x + + is zero when i.e. when 8 x=- or 4 x=- Therefore the zeroes of Sum of zeroes = -8+(-4)=-12 Product of the zeroes= POLYNOMIALS ( ) 3 2 f x x = + now we get different values of y corresponding to the different values of x. points A(0,2), B(1,5) and C(2,8) are plotted on the graph paper on a suitable scale. The points are joint together to get the graph of the given polynomial. Since the polynomial is linear we get the graph as a straight line. BETWEEN ZEROES AND COEFFICIENTS OF A POLYNOMIAL Find the zeroes of the quadratic polynomial 2 12 32 x x + + and verify the relationship between the zeroes and the coefficients. The given quadratic polynomial can be factorised as 2 12 32 ( 8)( 4) x x x x + + = + + is zero when ( 8) 0 x+ = or ( 4) 0 x+ = Therefore the zeroes of 2 12 32 x x + + are -8 and -4. 12 Product of the zeroes=-8 x -4=32 now we get different values of y corresponding to the different values of x. points A(0,2), B(1,5) and C(2,8) are plotted on the graph paper on a suitable scale. BETWEEN ZEROES AND COEFFICIENTS OF A POLYNOMIAL and verify the relationship 12 32 ( 8)( 4) x x x x + + = + + 4. Page 2 POLYNOMIALS GRAPHS OF POLYNOMIALS 1) Draw a graph of the polynomial f x x Solution: Let y = 3 2 x+ now we get different values of y corresponding to the different values of x. x 0 1 2 y 2 5 8 The points A(0,2), B(1,5) and C(2,8) are plotted on the graph paper on a suitable scale. The points are joint together to get the graph of the given polynomial. Since the polynomial is linear we get the graph as a straight line. RELATIONSHIP BETWEEN ZEROES AND COEFFICIENTS OF A POLYNOMIAL 2) Find the zeroes of the quadratic polynomial between the zeroes and the coefficients. Solution: The given quadratic polynomial can be Thus the value of 2 12 32 x x + + is zero when i.e. when 8 x=- or 4 x=- Therefore the zeroes of Sum of zeroes = -8+(-4)=-12 Product of the zeroes= POLYNOMIALS ( ) 3 2 f x x = + now we get different values of y corresponding to the different values of x. points A(0,2), B(1,5) and C(2,8) are plotted on the graph paper on a suitable scale. The points are joint together to get the graph of the given polynomial. Since the polynomial is linear we get the graph as a straight line. BETWEEN ZEROES AND COEFFICIENTS OF A POLYNOMIAL Find the zeroes of the quadratic polynomial 2 12 32 x x + + and verify the relationship between the zeroes and the coefficients. The given quadratic polynomial can be factorised as 2 12 32 ( 8)( 4) x x x x + + = + + is zero when ( 8) 0 x+ = or ( 4) 0 x+ = Therefore the zeroes of 2 12 32 x x + + are -8 and -4. 12 Product of the zeroes=-8 x -4=32 now we get different values of y corresponding to the different values of x. points A(0,2), B(1,5) and C(2,8) are plotted on the graph paper on a suitable scale. BETWEEN ZEROES AND COEFFICIENTS OF A POLYNOMIAL and verify the relationship 12 32 ( 8)( 4) x x x x + + = + + 4. POLYNOMIALS ( ) ( ) 2 Coefficient of x Coefficient of x - = (12) 1 - = =- ( ( Coefficient of x Sum of zeroes= Product of the zeroes= Coefficient of x - ? 3) Find the zeroes of the quadratic polynomial between the zeroes and the coefficients. Solution: Using the identity 2 2 ( )( ) a b a b a b - = + - ? v6 Hence, the value of v6 is zero when Hence x = -v6 and x = v6 are the zeroes of Sum of zeroes 6 6 0 = - = Product of zeroes ( ) ( ) 2 Coefficient of x Coefficient of x - 0 0 1 = = as the quadratic polynomial x 2 Constant term Coefficient of x 6 6 1 - = =- ( ( Coefficient of x Sum of zeroes= Product of the zeroes= Coefficient of x - ? 4) Find a quadratic polynomial, the sum and prod Solution; Let the quadratic polynomial be ax bx c Given that 6 a ß + =- b a - = b a ? = and x 5 c a a ß = = 5 c a ? = If a=1, then b=6 and c=5 Thus, quadratic polynomial which satisfies the given condition is POLYNOMIALS (12) 12 = =- 2 Constant term Coefficient of x 32 32 1 = = ) ) 2 Coefficient of x Constant term Sum of zeroes= Product of the zeroes= Coefficient of x Coefficient of x Find the zeroes of the quadratic polynomial 2 6 x - and verify the relationship between the zeroes and the coefficients. ( )( ) a b a b a b - = + - we can write, v6 v6 is zero when 6 0 x+ = or 6 0 x- = i.e. x=- 6 are the zeroes of the quadratic polynomial x 2 - 6 Product of zeroes 6 x - 6 6 = =- as the quadratic polynomial x 2 – 6 we can write it as x ) ) 2 Coefficient of x Constant term Sum of zeroes= Product of the zeroes= Coefficient of x Coefficient of x Find a quadratic polynomial, the sum and product of whose zeroes are -6 and 5 2 ax bx c + + and its zeroes be a and ß 6 b a ? = 5 c a ? = quadratic polynomial which satisfies the given condition is 2 6 5 x x + + 2 Constant term Coefficient of x 6 x=- or 6 x= write it as x 2 + 0x - 6 2 Constant term Coefficient of x 6 and 5 respectively 6 5 Page 3 POLYNOMIALS GRAPHS OF POLYNOMIALS 1) Draw a graph of the polynomial f x x Solution: Let y = 3 2 x+ now we get different values of y corresponding to the different values of x. x 0 1 2 y 2 5 8 The points A(0,2), B(1,5) and C(2,8) are plotted on the graph paper on a suitable scale. The points are joint together to get the graph of the given polynomial. Since the polynomial is linear we get the graph as a straight line. RELATIONSHIP BETWEEN ZEROES AND COEFFICIENTS OF A POLYNOMIAL 2) Find the zeroes of the quadratic polynomial between the zeroes and the coefficients. Solution: The given quadratic polynomial can be Thus the value of 2 12 32 x x + + is zero when i.e. when 8 x=- or 4 x=- Therefore the zeroes of Sum of zeroes = -8+(-4)=-12 Product of the zeroes= POLYNOMIALS ( ) 3 2 f x x = + now we get different values of y corresponding to the different values of x. points A(0,2), B(1,5) and C(2,8) are plotted on the graph paper on a suitable scale. The points are joint together to get the graph of the given polynomial. Since the polynomial is linear we get the graph as a straight line. BETWEEN ZEROES AND COEFFICIENTS OF A POLYNOMIAL Find the zeroes of the quadratic polynomial 2 12 32 x x + + and verify the relationship between the zeroes and the coefficients. The given quadratic polynomial can be factorised as 2 12 32 ( 8)( 4) x x x x + + = + + is zero when ( 8) 0 x+ = or ( 4) 0 x+ = Therefore the zeroes of 2 12 32 x x + + are -8 and -4. 12 Product of the zeroes=-8 x -4=32 now we get different values of y corresponding to the different values of x. points A(0,2), B(1,5) and C(2,8) are plotted on the graph paper on a suitable scale. BETWEEN ZEROES AND COEFFICIENTS OF A POLYNOMIAL and verify the relationship 12 32 ( 8)( 4) x x x x + + = + + 4. POLYNOMIALS ( ) ( ) 2 Coefficient of x Coefficient of x - = (12) 1 - = =- ( ( Coefficient of x Sum of zeroes= Product of the zeroes= Coefficient of x - ? 3) Find the zeroes of the quadratic polynomial between the zeroes and the coefficients. Solution: Using the identity 2 2 ( )( ) a b a b a b - = + - ? v6 Hence, the value of v6 is zero when Hence x = -v6 and x = v6 are the zeroes of Sum of zeroes 6 6 0 = - = Product of zeroes ( ) ( ) 2 Coefficient of x Coefficient of x - 0 0 1 = = as the quadratic polynomial x 2 Constant term Coefficient of x 6 6 1 - = =- ( ( Coefficient of x Sum of zeroes= Product of the zeroes= Coefficient of x - ? 4) Find a quadratic polynomial, the sum and prod Solution; Let the quadratic polynomial be ax bx c Given that 6 a ß + =- b a - = b a ? = and x 5 c a a ß = = 5 c a ? = If a=1, then b=6 and c=5 Thus, quadratic polynomial which satisfies the given condition is POLYNOMIALS (12) 12 = =- 2 Constant term Coefficient of x 32 32 1 = = ) ) 2 Coefficient of x Constant term Sum of zeroes= Product of the zeroes= Coefficient of x Coefficient of x Find the zeroes of the quadratic polynomial 2 6 x - and verify the relationship between the zeroes and the coefficients. ( )( ) a b a b a b - = + - we can write, v6 v6 is zero when 6 0 x+ = or 6 0 x- = i.e. x=- 6 are the zeroes of the quadratic polynomial x 2 - 6 Product of zeroes 6 x - 6 6 = =- as the quadratic polynomial x 2 – 6 we can write it as x ) ) 2 Coefficient of x Constant term Sum of zeroes= Product of the zeroes= Coefficient of x Coefficient of x Find a quadratic polynomial, the sum and product of whose zeroes are -6 and 5 2 ax bx c + + and its zeroes be a and ß 6 b a ? = 5 c a ? = quadratic polynomial which satisfies the given condition is 2 6 5 x x + + 2 Constant term Coefficient of x 6 x=- or 6 x= write it as x 2 + 0x - 6 2 Constant term Coefficient of x 6 and 5 respectively 6 5 POLYNOMIALS 5) Verify that 2,-3 and 6 are the zeroes of the cubic polynomial and then verify the relationship between the zeroes and the coefficients. Solution: Compare the given polynomial with a=1 b=-5 c=-12 and d=36. Now to verify if 2,-3 and 6 are the roots of the polynomial. i.e. we have to show that 2 ? 3 2 (2) 2 5(2) 12(2) 36 8 20 24 36 0 f = - - + = - - + = 3 2 ( 3) ( 3) 5( 3) 12( 3) 36 27 45 36 36 0 f - = - - - - - + =- - + + = 3 2 (6) (6) 5(6) 12(6) 36 216 180 72 36 0 f = - - + = - - + = Therefore 2,-3 and 6 are the zeroes of the cubic polynomial Let =2 a 3 ß =- and 6 ? = + 2 ( 3) 6 2 3 6 5 a ß ? + = + - + = - + = = 2( 3) ( 3)6 6(2) 6 18 12 12 a a ß ß? ? + + = - + - + =- - + =- = 2( 3)6 36 d a a ß? - = - =- = 6) If a and ß are the zeroes of the polynomial Solution: We know that ( ( Sum of zeroes= Coefficient of x - Product of the zeroes= Coefficient of x And hence, 1 1 a ß aß + = ? POLYNOMIALS 3 and 6 are the zeroes of the cubic polynomial 3 2 ( ) 5 12 36 f x x x x = - - + elationship between the zeroes and the coefficients. Compare the given polynomial with 3 2 ax bx cx d + + + we get, are the roots of the polynomial. 0 , 3 = 0 and 6 = 0 (2) 2 5(2) 12(2) 36 8 20 24 36 0 = - - + = - - + = ( 3) ( 3) 5( 3) 12( 3) 36 27 45 36 36 0 - = - - - - - + =- - + + = (6) (6) 5(6) 12(6) 36 216 180 72 36 0 = - - + = - - + = 3 and 6 are the zeroes of the cubic polynomial 3 2 5 12 36 x x x - - + + 2 ( 3) 6 2 3 6 5 b a - + = + - + = - + = = 2( 3) ( 3)6 6(2) 6 18 12 12 c a + + = - + - + =- - + =- = are the zeroes of the polynomial 2 ( ) f x ax bx c = + + then find 1 1 a + ( ) ) 2 Coefficient of x Coefficient of x - i.e. b a a ß - + = 2 Constant term Coefficient of x i.e. c a a ß = a ß a ß aß + + = b a c a - = b c - = ? 1 1 b c a ß - + = ( ) 5 12 36 f x x x x = - - + 5 12 36 1 1 ß + Page 4 POLYNOMIALS GRAPHS OF POLYNOMIALS 1) Draw a graph of the polynomial f x x Solution: Let y = 3 2 x+ now we get different values of y corresponding to the different values of x. x 0 1 2 y 2 5 8 The points A(0,2), B(1,5) and C(2,8) are plotted on the graph paper on a suitable scale. The points are joint together to get the graph of the given polynomial. Since the polynomial is linear we get the graph as a straight line. RELATIONSHIP BETWEEN ZEROES AND COEFFICIENTS OF A POLYNOMIAL 2) Find the zeroes of the quadratic polynomial between the zeroes and the coefficients. Solution: The given quadratic polynomial can be Thus the value of 2 12 32 x x + + is zero when i.e. when 8 x=- or 4 x=- Therefore the zeroes of Sum of zeroes = -8+(-4)=-12 Product of the zeroes= POLYNOMIALS ( ) 3 2 f x x = + now we get different values of y corresponding to the different values of x. points A(0,2), B(1,5) and C(2,8) are plotted on the graph paper on a suitable scale. The points are joint together to get the graph of the given polynomial. Since the polynomial is linear we get the graph as a straight line. BETWEEN ZEROES AND COEFFICIENTS OF A POLYNOMIAL Find the zeroes of the quadratic polynomial 2 12 32 x x + + and verify the relationship between the zeroes and the coefficients. The given quadratic polynomial can be factorised as 2 12 32 ( 8)( 4) x x x x + + = + + is zero when ( 8) 0 x+ = or ( 4) 0 x+ = Therefore the zeroes of 2 12 32 x x + + are -8 and -4. 12 Product of the zeroes=-8 x -4=32 now we get different values of y corresponding to the different values of x. points A(0,2), B(1,5) and C(2,8) are plotted on the graph paper on a suitable scale. BETWEEN ZEROES AND COEFFICIENTS OF A POLYNOMIAL and verify the relationship 12 32 ( 8)( 4) x x x x + + = + + 4. POLYNOMIALS ( ) ( ) 2 Coefficient of x Coefficient of x - = (12) 1 - = =- ( ( Coefficient of x Sum of zeroes= Product of the zeroes= Coefficient of x - ? 3) Find the zeroes of the quadratic polynomial between the zeroes and the coefficients. Solution: Using the identity 2 2 ( )( ) a b a b a b - = + - ? v6 Hence, the value of v6 is zero when Hence x = -v6 and x = v6 are the zeroes of Sum of zeroes 6 6 0 = - = Product of zeroes ( ) ( ) 2 Coefficient of x Coefficient of x - 0 0 1 = = as the quadratic polynomial x 2 Constant term Coefficient of x 6 6 1 - = =- ( ( Coefficient of x Sum of zeroes= Product of the zeroes= Coefficient of x - ? 4) Find a quadratic polynomial, the sum and prod Solution; Let the quadratic polynomial be ax bx c Given that 6 a ß + =- b a - = b a ? = and x 5 c a a ß = = 5 c a ? = If a=1, then b=6 and c=5 Thus, quadratic polynomial which satisfies the given condition is POLYNOMIALS (12) 12 = =- 2 Constant term Coefficient of x 32 32 1 = = ) ) 2 Coefficient of x Constant term Sum of zeroes= Product of the zeroes= Coefficient of x Coefficient of x Find the zeroes of the quadratic polynomial 2 6 x - and verify the relationship between the zeroes and the coefficients. ( )( ) a b a b a b - = + - we can write, v6 v6 is zero when 6 0 x+ = or 6 0 x- = i.e. x=- 6 are the zeroes of the quadratic polynomial x 2 - 6 Product of zeroes 6 x - 6 6 = =- as the quadratic polynomial x 2 – 6 we can write it as x ) ) 2 Coefficient of x Constant term Sum of zeroes= Product of the zeroes= Coefficient of x Coefficient of x Find a quadratic polynomial, the sum and product of whose zeroes are -6 and 5 2 ax bx c + + and its zeroes be a and ß 6 b a ? = 5 c a ? = quadratic polynomial which satisfies the given condition is 2 6 5 x x + + 2 Constant term Coefficient of x 6 x=- or 6 x= write it as x 2 + 0x - 6 2 Constant term Coefficient of x 6 and 5 respectively 6 5 POLYNOMIALS 5) Verify that 2,-3 and 6 are the zeroes of the cubic polynomial and then verify the relationship between the zeroes and the coefficients. Solution: Compare the given polynomial with a=1 b=-5 c=-12 and d=36. Now to verify if 2,-3 and 6 are the roots of the polynomial. i.e. we have to show that 2 ? 3 2 (2) 2 5(2) 12(2) 36 8 20 24 36 0 f = - - + = - - + = 3 2 ( 3) ( 3) 5( 3) 12( 3) 36 27 45 36 36 0 f - = - - - - - + =- - + + = 3 2 (6) (6) 5(6) 12(6) 36 216 180 72 36 0 f = - - + = - - + = Therefore 2,-3 and 6 are the zeroes of the cubic polynomial Let =2 a 3 ß =- and 6 ? = + 2 ( 3) 6 2 3 6 5 a ß ? + = + - + = - + = = 2( 3) ( 3)6 6(2) 6 18 12 12 a a ß ß? ? + + = - + - + =- - + =- = 2( 3)6 36 d a a ß? - = - =- = 6) If a and ß are the zeroes of the polynomial Solution: We know that ( ( Sum of zeroes= Coefficient of x - Product of the zeroes= Coefficient of x And hence, 1 1 a ß aß + = ? POLYNOMIALS 3 and 6 are the zeroes of the cubic polynomial 3 2 ( ) 5 12 36 f x x x x = - - + elationship between the zeroes and the coefficients. Compare the given polynomial with 3 2 ax bx cx d + + + we get, are the roots of the polynomial. 0 , 3 = 0 and 6 = 0 (2) 2 5(2) 12(2) 36 8 20 24 36 0 = - - + = - - + = ( 3) ( 3) 5( 3) 12( 3) 36 27 45 36 36 0 - = - - - - - + =- - + + = (6) (6) 5(6) 12(6) 36 216 180 72 36 0 = - - + = - - + = 3 and 6 are the zeroes of the cubic polynomial 3 2 5 12 36 x x x - - + + 2 ( 3) 6 2 3 6 5 b a - + = + - + = - + = = 2( 3) ( 3)6 6(2) 6 18 12 12 c a + + = - + - + =- - + =- = are the zeroes of the polynomial 2 ( ) f x ax bx c = + + then find 1 1 a + ( ) ) 2 Coefficient of x Coefficient of x - i.e. b a a ß - + = 2 Constant term Coefficient of x i.e. c a a ß = a ß a ß aß + + = b a c a - = b c - = ? 1 1 b c a ß - + = ( ) 5 12 36 f x x x x = - - + 5 12 36 1 1 ß + POLYNOMIALS 7) Find the zeroes of the polynomial the coefficients Solution: Given that, 2 ( ) 4 5 4 3 5 f x x x = + - 2 ( ) 4 5 10 6 3 5 f x x x x = + - - ( )( ) ( ) 2 5 2 5 3 f x x x = + - The zeroes of the polynomial are given by i.e. ( )( ) 2 5 2 5 3 0 x x + - = ? 2 5 0 x+ = or 2 5 3 0 x- = 5 2 x - = or 3 2 5 x= Hence zeroes are 3 2 5 a = and a ß + = 3 5 3 5 6 10 4 1 2 2 2 5 2 5 4 5 4 5 5 ? ? + = - = = = ? ? ? ? ? ? 3 5 3 x 2 4 2 5 aß ? ? - - = = ? ? ? ? ? ? 4 1 4 5 5 b a - - - = = 3 5 3 4 5 c a - - = = b a a ß - + = c a a ß = ( ( Coefficient of x Sum of zeroes= Product of the zeroes= Coefficient of x - ? 8) If a and ß are the zeroes of the polynomial Solution: We know that ( ( Sum of zeroes= Coefficient of x - POLYNOMIALS Find the zeroes of the polynomial 2 ( ) 4 5 4 3 5 f x x x = + - and verify the relationship between the zeroes and The zeroes of the polynomial are given by ( ) 0 f x = 2 5 2 5 3 0 2 5 3 0 - = and 5 2 ß - = 3 5 3 5 6 10 4 1 2 2 2 5 2 5 4 5 4 5 5 - - - + = - = = = 3 5 3 4 - - = = ) ) 2 Coefficient of x Constant term Sum of zeroes= Product of the zeroes= Coefficient of x Coefficient of x are the zeroes of the polynomial 2 ( ) f x ax bx c = + + then find a ß ß a + ( ) ) 2 Coefficient of x Coefficient of x - i.e. b a a ß - + = POLYNOMIALS verify the relationship between the zeroes and 2 Constant term Coefficient of x a ß ß a + Page 5 POLYNOMIALS GRAPHS OF POLYNOMIALS 1) Draw a graph of the polynomial f x x Solution: Let y = 3 2 x+ now we get different values of y corresponding to the different values of x. x 0 1 2 y 2 5 8 The points A(0,2), B(1,5) and C(2,8) are plotted on the graph paper on a suitable scale. The points are joint together to get the graph of the given polynomial. Since the polynomial is linear we get the graph as a straight line. RELATIONSHIP BETWEEN ZEROES AND COEFFICIENTS OF A POLYNOMIAL 2) Find the zeroes of the quadratic polynomial between the zeroes and the coefficients. Solution: The given quadratic polynomial can be Thus the value of 2 12 32 x x + + is zero when i.e. when 8 x=- or 4 x=- Therefore the zeroes of Sum of zeroes = -8+(-4)=-12 Product of the zeroes= POLYNOMIALS ( ) 3 2 f x x = + now we get different values of y corresponding to the different values of x. points A(0,2), B(1,5) and C(2,8) are plotted on the graph paper on a suitable scale. The points are joint together to get the graph of the given polynomial. Since the polynomial is linear we get the graph as a straight line. BETWEEN ZEROES AND COEFFICIENTS OF A POLYNOMIAL Find the zeroes of the quadratic polynomial 2 12 32 x x + + and verify the relationship between the zeroes and the coefficients. The given quadratic polynomial can be factorised as 2 12 32 ( 8)( 4) x x x x + + = + + is zero when ( 8) 0 x+ = or ( 4) 0 x+ = Therefore the zeroes of 2 12 32 x x + + are -8 and -4. 12 Product of the zeroes=-8 x -4=32 now we get different values of y corresponding to the different values of x. points A(0,2), B(1,5) and C(2,8) are plotted on the graph paper on a suitable scale. BETWEEN ZEROES AND COEFFICIENTS OF A POLYNOMIAL and verify the relationship 12 32 ( 8)( 4) x x x x + + = + + 4. POLYNOMIALS ( ) ( ) 2 Coefficient of x Coefficient of x - = (12) 1 - = =- ( ( Coefficient of x Sum of zeroes= Product of the zeroes= Coefficient of x - ? 3) Find the zeroes of the quadratic polynomial between the zeroes and the coefficients. Solution: Using the identity 2 2 ( )( ) a b a b a b - = + - ? v6 Hence, the value of v6 is zero when Hence x = -v6 and x = v6 are the zeroes of Sum of zeroes 6 6 0 = - = Product of zeroes ( ) ( ) 2 Coefficient of x Coefficient of x - 0 0 1 = = as the quadratic polynomial x 2 Constant term Coefficient of x 6 6 1 - = =- ( ( Coefficient of x Sum of zeroes= Product of the zeroes= Coefficient of x - ? 4) Find a quadratic polynomial, the sum and prod Solution; Let the quadratic polynomial be ax bx c Given that 6 a ß + =- b a - = b a ? = and x 5 c a a ß = = 5 c a ? = If a=1, then b=6 and c=5 Thus, quadratic polynomial which satisfies the given condition is POLYNOMIALS (12) 12 = =- 2 Constant term Coefficient of x 32 32 1 = = ) ) 2 Coefficient of x Constant term Sum of zeroes= Product of the zeroes= Coefficient of x Coefficient of x Find the zeroes of the quadratic polynomial 2 6 x - and verify the relationship between the zeroes and the coefficients. ( )( ) a b a b a b - = + - we can write, v6 v6 is zero when 6 0 x+ = or 6 0 x- = i.e. x=- 6 are the zeroes of the quadratic polynomial x 2 - 6 Product of zeroes 6 x - 6 6 = =- as the quadratic polynomial x 2 – 6 we can write it as x ) ) 2 Coefficient of x Constant term Sum of zeroes= Product of the zeroes= Coefficient of x Coefficient of x Find a quadratic polynomial, the sum and product of whose zeroes are -6 and 5 2 ax bx c + + and its zeroes be a and ß 6 b a ? = 5 c a ? = quadratic polynomial which satisfies the given condition is 2 6 5 x x + + 2 Constant term Coefficient of x 6 x=- or 6 x= write it as x 2 + 0x - 6 2 Constant term Coefficient of x 6 and 5 respectively 6 5 POLYNOMIALS 5) Verify that 2,-3 and 6 are the zeroes of the cubic polynomial and then verify the relationship between the zeroes and the coefficients. Solution: Compare the given polynomial with a=1 b=-5 c=-12 and d=36. Now to verify if 2,-3 and 6 are the roots of the polynomial. i.e. we have to show that 2 ? 3 2 (2) 2 5(2) 12(2) 36 8 20 24 36 0 f = - - + = - - + = 3 2 ( 3) ( 3) 5( 3) 12( 3) 36 27 45 36 36 0 f - = - - - - - + =- - + + = 3 2 (6) (6) 5(6) 12(6) 36 216 180 72 36 0 f = - - + = - - + = Therefore 2,-3 and 6 are the zeroes of the cubic polynomial Let =2 a 3 ß =- and 6 ? = + 2 ( 3) 6 2 3 6 5 a ß ? + = + - + = - + = = 2( 3) ( 3)6 6(2) 6 18 12 12 a a ß ß? ? + + = - + - + =- - + =- = 2( 3)6 36 d a a ß? - = - =- = 6) If a and ß are the zeroes of the polynomial Solution: We know that ( ( Sum of zeroes= Coefficient of x - Product of the zeroes= Coefficient of x And hence, 1 1 a ß aß + = ? POLYNOMIALS 3 and 6 are the zeroes of the cubic polynomial 3 2 ( ) 5 12 36 f x x x x = - - + elationship between the zeroes and the coefficients. Compare the given polynomial with 3 2 ax bx cx d + + + we get, are the roots of the polynomial. 0 , 3 = 0 and 6 = 0 (2) 2 5(2) 12(2) 36 8 20 24 36 0 = - - + = - - + = ( 3) ( 3) 5( 3) 12( 3) 36 27 45 36 36 0 - = - - - - - + =- - + + = (6) (6) 5(6) 12(6) 36 216 180 72 36 0 = - - + = - - + = 3 and 6 are the zeroes of the cubic polynomial 3 2 5 12 36 x x x - - + + 2 ( 3) 6 2 3 6 5 b a - + = + - + = - + = = 2( 3) ( 3)6 6(2) 6 18 12 12 c a + + = - + - + =- - + =- = are the zeroes of the polynomial 2 ( ) f x ax bx c = + + then find 1 1 a + ( ) ) 2 Coefficient of x Coefficient of x - i.e. b a a ß - + = 2 Constant term Coefficient of x i.e. c a a ß = a ß a ß aß + + = b a c a - = b c - = ? 1 1 b c a ß - + = ( ) 5 12 36 f x x x x = - - + 5 12 36 1 1 ß + POLYNOMIALS 7) Find the zeroes of the polynomial the coefficients Solution: Given that, 2 ( ) 4 5 4 3 5 f x x x = + - 2 ( ) 4 5 10 6 3 5 f x x x x = + - - ( )( ) ( ) 2 5 2 5 3 f x x x = + - The zeroes of the polynomial are given by i.e. ( )( ) 2 5 2 5 3 0 x x + - = ? 2 5 0 x+ = or 2 5 3 0 x- = 5 2 x - = or 3 2 5 x= Hence zeroes are 3 2 5 a = and a ß + = 3 5 3 5 6 10 4 1 2 2 2 5 2 5 4 5 4 5 5 ? ? + = - = = = ? ? ? ? ? ? 3 5 3 x 2 4 2 5 aß ? ? - - = = ? ? ? ? ? ? 4 1 4 5 5 b a - - - = = 3 5 3 4 5 c a - - = = b a a ß - + = c a a ß = ( ( Coefficient of x Sum of zeroes= Product of the zeroes= Coefficient of x - ? 8) If a and ß are the zeroes of the polynomial Solution: We know that ( ( Sum of zeroes= Coefficient of x - POLYNOMIALS Find the zeroes of the polynomial 2 ( ) 4 5 4 3 5 f x x x = + - and verify the relationship between the zeroes and The zeroes of the polynomial are given by ( ) 0 f x = 2 5 2 5 3 0 2 5 3 0 - = and 5 2 ß - = 3 5 3 5 6 10 4 1 2 2 2 5 2 5 4 5 4 5 5 - - - + = - = = = 3 5 3 4 - - = = ) ) 2 Coefficient of x Constant term Sum of zeroes= Product of the zeroes= Coefficient of x Coefficient of x are the zeroes of the polynomial 2 ( ) f x ax bx c = + + then find a ß ß a + ( ) ) 2 Coefficient of x Coefficient of x - i.e. b a a ß - + = POLYNOMIALS verify the relationship between the zeroes and 2 Constant term Coefficient of x a ß ß a + POLYNOMIALS Constant term Product of the zeroes= Coefficient of x ( ) 2 2 2 2 a ß aß a ß a ß ß a aß aß + - + + = = ? 2 2 b ac ac a ß ß a - + == 9) If a and ß are the zeroes of the polynomial then find the value of k Solution: Since a and ß are zeroes of the polynomial. a +ß = = -6, aß = = k and a ß We know that, ( ) 2 2 4 a ß aß a ß + - = - Substituting the values we get, (-6) 2 – 4k = 4 2 36 – 4k = 16 36 – 16 = 4k 20 = 4k Therefore k = 5 10) Find the zeroes of the polynomial relationship between the zeroes and coefficients. Solution: We have ( ) 2 2 2 2 ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) f x abx b ac x bc f x abx b x acx bc f x bx ax b c ax b f x ax b bx c = + - - = + - - = + - + = + - POLYNOMIALS 2 Constant term Coefficient of x i.e. c a a ß = 2 a ß aß 2 2 2 2 2 2 2 2 b c b c b ac a a a a a c c c a a a - ? ? ? ? - - - ? ? ? ? ? ? ? ? = = = = es of the polynomial 2 ( ) 6 f x x x k = + + such that a ß - = are zeroes of the polynomial. 4 a ß - = ( ) 2 2 a ß aß a ß + - = - Find the zeroes of the polynomial ( ) 2 2 ( ) f x abx b ac x bc = + - - and verify the elationship between the zeroes and coefficients. f x abx b ac x bc 2 2 b ac ac - = = = = 4 a ß - = and verify theRead More

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