, a 1. , a 2. ,..., a n


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1 CHAPTER Points to Remember :. Let x be a variable, n be a positive integer and a 0, a, a,..., a n be constants. Then n f ( x) a x a x... a x a, is called a polynomial in variable x. n n n 0 POLNOMIALS. The exponent of the highest degree term in a polynomial is known as its degree.. Degree Name of Polynomial Form of the Polynomial 0 Constant Polynomial f(x) = a, a is constant Linear Polynomial f(x) = ax + b, a 0 Quadratic Polynomial f(x) = ax + bx + c; a 0 Cubic Polynomial f(x) = ax + bx + cx + d; a 0. If f(x) is a polynomial and is any real number, then the real number obtained by replacing x by in f(x) at x = and is denoted by f( ).. A real number is a zero of a polynomial f(x), if f( ) = A polynomial of degree n can have at most n real zeroes. 7. Geometrically, the zeroes of a polynomial f(x) are the xcoordinates of the points where the graph y = f(x) intersects xaxis. 8. For any quadratic polynomial ax + bx + c = 0, a 0, the graph of the corresponding equation y = ax + bx + c has one of the two shapes either open upwards like or downwards like, depending on whether a > 0 or a < 0. These curves are called Parabolas. 9. If and are the zeroes of a quadratic polynomial f(x) = ax + bx + c, a 0 then b coefficient of x a coefficient of x c constant term a coefficient of x 0. If are the zeroes of a cubic polynomial f(x) = ax + bx + cx + d, a 0 then b coefficient of x a coefficient of x c coefficient of x a coefficient of x d constant term a coefficient of x. Division Algorithm : If f(x) is a polynomial and g(x) is a nonzero polynomial, then there exist two polynomials q(x) and r(x) such that f ( x) g( x) q( x) r( x), where r(x) = 0 or degree of r(x) < degree of g(x). POLNOMIALS MATHEMATICS
2 ILLUSTRATIVE EAMPLES Example. Draw the graph of the polynomial f(x) = x x. Obtain the vertex of this parabola. Also, read the zeroes of the polynomial, if possible from the graph. Solution. Let y = f(x) = x x. The following table gives the values of y for various values of x. x y x x After plotting the points in the table on a graph paper, draw a freehand continuous curve (parabola) through all the points plotted. y x (, ) (, ) (0, ) (, 0) (, 0) (, ) (, ) (, ) (, ) y Vertex of Parabola : On comparing the polynomial x x with ax + bx + c, we get a =, b D b =, c =. The vertex of the parabola has the coordinates, a a, where D = b ac. Now, D = b ac = ( ) () ( )= + = 6 ( ) 6 Coordinate of vertex is, (, ) () () Zeroes of f(x) = x x Since the parabola f(x) = x x cuts the xaxis at (, 0) and (, 0). zeroes of f(x) are and. MATHEMATICS POLNOMIALS x
3 Example. Find the zeroes of the quadratic polynomial x x 8, and verify the relationship between the zeroes and their coefficients. Solution. Let f(x) = x x 8 The zeroes of f(x) are given by f(x) = 0. x x 8 = 0 x x + x 8 = 0 x (x ) + (x ) = 0 (x + ) (x ) = 0 x = or x = Ans. ( ) coefficient of x Verification : Here, sum of zeroes = + ( ) = = coefficient of x 8 constant term and, product of zeroes ( ) 8 coefficient of x Example. Find a quadratic polynomial whose sum and product of the zeroes is Solution. and respectively. We know that a quadratic polynomial when the sum and product of its zeroes are given by f(x) = k {x (sum of the zeroes) x + Product of the zeroes}, where k is a constant. Required quadratic polynomial f(x) is given by f ( x) k x x OR Let the required quadratic polynomial be ax + bx + c. b b Since, sum of zeroes a a and, product of zeroes c c c a a a If a = ; then b = and c =. Required quadratic polynomial = ax + bx + c = x x POLNOMIALS MATHEMATICS
4 Example. Divide the polynomial f(x) = x x + x + by the polynomial x + x and verify the division algorithm. Solution. On writing the dividend and the divisor in the standard form, we get : Example. Dividend p(x) = x x + x + and divisor g(x) = x x +. Now, x x + x + 0x x x + + xx+ + x + x x x + x + x x + x x x + x + x + x + x + x Clearly, quotient = x + x and remainder = 8. Verification of Division Algorithm : Quotient Divisor + Remainder = (x + x ) (x x + ) + 8 = x x + x + x x + x x + x + 8 = x x + x + = Dividend Hence verified. Obtain all the zeroes of the polynomial f ( x) x + 6x x 0x, if two of its zeroes are and. [NCERT] Solution. Since and are two zeroes of f(x). x x x ( x ) is a factor of f(x). Also, x is a factor of f (x). Let us now divide f(x) by x. We have, x x + 6 x x 0 x x + x + x + 0 x x + 6 x + x 0x 6 x + 0 x 0x + x x MATHEMATICS POLNOMIALS
5 By division algorithm, we have x + 6x x 0x = (x ) (x + x + ) ( x ) ( x ) ( x ) Hence, the zeroes of f(x) are,, and. PRACTICE EERCISE. Without drawing actual graph, find the zeroes of the polynomials if any. Give reason. (a) x x 8 (b) x x + (c) x + x + (d) x (e) x + x + (f) x + x. The graphs of y = p(x) are given below, for some polynomials p(x). Find the number of zeroes of p(x), in each case. (a) (b) (c) (d) (e) (f) 6 POLNOMIALS MATHEMATICS
6 . The graph of y = p(x) are given below, for some polynomial p(x). Find the zeroes of the corresponding polynomial. (, 7) 8 6 (, 0) (, 0) (0, 8) 0 (, 9) (a) (, ) y = x x 8 (, 7) (, 0) (, 0) 0 (c) (, ) y = x x 8 8 (0, 6) 6 MATHEMATICS POLNOMIALS 7 0 (, 0) (, 0) (, 6) (, ) 6 8 (b) (, 6) y= x + x+6 (, ) (, 6) 0 (d). Draw the graph of each of the following polynomials and if possible, read the zero(s) from the graph : (a) x x + 9 (b) x + x (c) x + x (d) x 8x + 6 (e) x (f) x x. Draw the graph of the polynomial x x 0. Read off the zeroes of the polynomial from the graph. Also show the axis of symmetry on it. 6. Show that and are the zeroes of the polynomial p(x) = x x. 7. Show that the polynomial p(x) = x x + 9 have no real zeroes. 8
7 8. Find the zeroes of each of the following quadratic polynomial. Also, in each case, verify the relationship between the zeroes and its coefficients : (a) x + 8x + (b) x + x (c) x 7x + 0 (d) y (e) u (f) t 9. Find a quadratic polynomial each with the given numbers as the sum and the product of its zeroes respectively : (a) and (b) and (c) and 0 (d) and 0. Verify that the numbers given alongside of the cubic polynomials below are their zeroes. Also, verify the relationship between the zeroes and the coefficients in each case. (a) x x ; 0, and (b) y 9y ; 0, and (c) x x + x + ;, and (d) x x x ;, and (e) x x + x ;, and (f) 6y + y y ;, and. Find a cubic polynomial with the sum, sum of product of its zeroes taken two at a time and the product of its zeroes respectively as given below : (a), 7 and 0 (b), and 8 (c), and 0. Write a cubic polynomial with zeroes :,, and hence find : (a) sum of its zeroes (b) sum of its zeroes taking any two at a time (c) product of three zeroes. Apply the division algorithm to find the quotient and the remainder on division of p(x) by g(x) as given below : (a) p(x) = x + x + 9x, g(x) = + x (b) p(x) = 6x + x 9x 6, g(x) = x + x (c) p(x) = x x + 6, g(x) = x (d) p(x)= x + x + x +, g(x) = + x + x (e) p(x) = x 7x + x + 6, g(x) = x (f) p(x) = 6x + 9x + 8x, g(x) = x + (g) p(x) = x x x 7x +, g(x) = x x + (h) p(x) = 6x + x x + x +, g(x) = x x +. Check whether the first polynomial is a factor of the second polynomial by applying the division algorithm : (a) x + 8; x + x + 6x (b) x ; x x + x 9 (c) x ; x x + x (d) y y + ; y + y 9y 6y + 0 (e) x + x + ; x + 7x + 0x + x + 8 (f) x x + ; x x + x + x +. Obtain all the zeroes of the polynomial f(x) = x x x + 9x 6; if two of its zeroes are and. 6. Obtain all the zeroes of the polynomial x x 7x + x + 6 ; if two of its zeroes are 8 POLNOMIALS MATHEMATICS.
8 7. Find the value of a and b so that, are the zeroes of the polynomial x + 0x + ax + b. 8. On dividing x x + x by a polynomial g(x), the quotient and remainder are x and 7x 9 respectively. Find g(x). 9. On dividing x x + 6 by a polynomial g(x), the quotient and remainder are x and x + 0 respectively. Find g(x). 0. What must be subtracted from 8x + x x + 7x 8 so that the resulting polynomial is exactly divisible by x + x?. What must be added to the polynomial p(x) = x + x x + x so that the resulting polynomial is exactly divisible by x + x?. If the polynomial x 6x + 6x 6x + 0 a is divided by another polynomial x x + k, the remainder comes out to be x + a. Find k and a. HINTS TO SELECTED QUESTIONS 7. Graphically, we observe that the given curve do not intersect xaxis. So, given polynomial do not have real roots. OR, here d = b ac = ( ) () (9) = 6 6 = 0 < 0 given polynomial has no real roots. 0. Dividing 8x + x x + 7x 8 by x + x, we get Quotient = x + x and Remainder = x 0. Clearly, polynomial to be subtracted = Remainder = x 0.. here, required polynomial to be added = Remainder. By division algorithm, we have f(x) (x + a) = x 6x + 6x 6x + 0 a is exactly divisible by x x + k. x x + k ) x 6 x + 6 x 6 x + 0 a ( x x + (8 k) x x + kx + x + (6 k) x 6 x x + 8 x kx kx Accoding to given conditions, we must have 0 + k = 0 and 0 a 8k + k = 0 k = and a =. (8 k) x (6 k) x + 0 a (8 k) x (6 k) x + (8 k k ) + ( 0 + k) x + (0 a 8 k + k ) MATHEMATICS POLNOMIALS 9
9 MULTIPLE CHOICE QUESTIONS Mark the correct alternative in each of the following :. The value of p(x) = x 6x + x 6 at x = is : (a) 8 (b) 0 (c) (d) none of these. The graph of y = x x 8 cuts the xaxis at: (a) and (b) and (c) do not cut (d) none of these. The graph of y = x + x cuts the xaxis at : (a) and 0 (b) and (c) do not cut (d) none of these. The zeroes of y = x + 7x + are : (a) and (b) and (c) and (d) and. The zeroes of f ( x) x x are : (a) and (b) and (c) and (d) none of these 6. The sum of squares of zeroes of f(x) = x 8x + k is 0. the value of k is (a) (b) (c) (d) none of these 7. If are the zeroes of f(x) = x + x +, then is : (a) 0 (b) (c) (d) none of these 8. The sum and product of zeroes of quadratic polynomial x 8x + is : 8 (a), (b) 8 8, (c), (d) none of these 9. The quadratic polynomial whose sum and product of zeroes is 7 and is: (a) x 7x + (b) x + 7x (c) x 7x (d) none of these 0. The cubic equation with zeroes, and is : (a) x 9x + 6 x (b) x + 9x 6x (c) x 9x 6x + (d) none of these. The quotient when f(x) = x 6x + x 6 is divided by x + is : (a) x + 8x 7 (b) x 8x + 7 (c) x 8x 7 (d) none of these. The remainder when f (x) = x + x + x + is divided by x + is (a) 6 (b) 8 (c) (d) none of these. The two zeroes of f(x) = x 6x 6x + 8x are ±, the other two zeroes are : (a) 7 and (b) 7 and (c) 7 and (d) none of these. If two zeroes of p(x)= x + x x 9x 6 are and, the other two zeroes are: (a) and (b) and (c) and (d) none of these. If the zeroes of f(x) = x x + x + are a b, a, a + b, then value of a and b is : (a) a, b (b) a, b (c) a, b (d) none of these 0 POLNOMIALS MATHEMATICS
10 VER SHORT ANSWER TPE QUESTIONS ( MARK QUESTIONS). The graph of polynomial y f ( x) are given below. Find the number of zeroes of the corresponding polynomial. O O MATHEMATICS POLNOMIALS ( i) ( ii) ( iii) O O ( iv) ( v) ( vi) O O. The graph of polynomial y f ( x) are given below. For each of the graphs, find the zeroes of the corresponding polynomial. 0 ( i) 0 ( ii)
11 0 ( iii). Find the zeroes of the polynomial 9x.. Write a polynomial whose zeroes are and.. How many maximum zeroes will the polynomial x + 6x 7 can have? 6. Write the sum and the product of the zeroes of the polynomial x x + 9. POLNOMIALS MATHEMATICS 0 7. Write a polynomial whose sum and product of zeroes are and 9 respectively. 8. Show that x = is a zero of the polynomial f(x) = x x + x. 9. If and are the zeroes of the polynomial x x + 6, find the value of. ( iv) 0. Give an example of polynomials p(x), g(x), q(x) and r(x), which satisfy the division algorithm and deg r(x) = 0.. What should be added to the polynomial p(x) = x x +, so that is a zero of p(x)?. What should be subtracted from the polynomial x 7x so that x = is a zero of the polynomial?. What is the value of p in the polynomial f ( x) x x p if is a zero of the polynomial?. How many zeroes does the polynomial p( x) x( x )( x ) have in all?. How many maximum number of zeroes a quadratic polynomial can have? 6. Find a quadratic polynomial whose two roots are and. 7. Find the common zero of x, x and ( x )? 8. For what value of x, both the polynomials 9. Find a cubic polynomial whose three zeroes are 0,,. p( x) x x 6 and q( x) x x becomes zero. 0. If are the zeroes of quadratic polynomial x x +, form a quadratic polynomial whose zeroes are and.
12 M.M : 0 General Instructions : PRACTICE TEST Q.  carry marks, Q. 8 carry marks and Q. 90 carry marks each.. The graph of y = p(x) are given below. find the number of zeroes of p(x). (a) (b) Time : hour. Find zeroes of f (x)= x + x. Also, verify the relationship between zeroes and its coefficients.. Find a quadratic polynomial whose sum and product of the zeroes are and respectively.. Using division algorithm, show that (x + ) is a factor of p(x) = x x 0.. Draw the graph of the polynomial p(x)= x + x. 6. On dividing x x + x + by a polynomial g(x), the quotient and remainder are (x ) and ( x + ) respectively. Find g(x). 7. If the zeroes of the polynomial x x + x + are a b, a, a + b, find a and b. 8. Divide p(x) = 9x x + by q(x) = x + x. Also, find quotient and remainder. 9. Find all the zeroes of the polynomial f(x) = x x x + 6x, if its two zeroes are and. 0. Verify that,, are the zeroes of the cubic polynomial p(x) = x x x, and then verify the relationship between the zeroes and the coefficients. ANSWERS OF PRACTICE EERCISE. (a) and (b) and (c) no zero (d) and (e) and (f) and. (a) (b) (c) (d) MATHEMATICS POLNOMIALS (e) (f). (a), (b), (c), 0, (d).,,. (a) No zero (b) 0 and (c) and (d) (e) 0 (f) 0 and. zeroes and ; Axis of symmetry; x
13 8. (a) and 6 (b) and (c) and (d) ± (e) ± (f) 9. (a) x x + (b) x x (c) x x (d) x x. (a) x + x 7 (b) x x x + (c) x 6x + 8. x x 6x + 8; (a) (b) 6 (c) 8. (a) quotient = 7x + x +, remainder = (b) quotient = 6x +, remainder = 8x 60 (c) quotient = x, remainder = x + 0 (d) quotient = x, remainder = 9x + 0 (e) quotient = x + x 6, remainder = (f) quotient = x + x +, remainder = 0 (g) quotient = x 9x 9, remainder = 0 x + 00 (h) quotient = x + x x, remainder = x +. (a) yes (b) yes (c) no (d) yes (e) no (f) no POLNOMIALS MATHEMATICS.,,, 6., 7. a = 7, b = 8 8. x 9. x + 0. x 0. x. k =, a = ANSWERS OF MULTIPLE CHOICE QUESTIONS. (b). (a). (c). (b). (d) 6. (b) 7. (c) 8. (b) 9. (c) 0. (a). (b). (d). (a). (c). (b) ANSWERS OF VER SHORT ANSWER TPE QUESTIONS. (i) (ii) (iii) (iv) (v) (vi). (i) and (ii), 0 and (iii), and (iv). 9.. x 7x sum =, product = 7. x x p(x) = x + x + 7, g(x) = x +, q(x) = x +, r(x) = x 6x + 7. x + 8. x = 9. x 9x 0. k (x + x + ) ANSWERS OF PRACTICE TEST. (a) (b). and 7. x x 6. x x + 7. a =, b 8. quotient = x x, remainder = x + 9.,,,
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