CHAPTER POLYNOMIALS (A) Main Concepts and Results Meaning of a Polynomial Degree of a polynomial Coefficients Monomials, Binomials etc. Constant, Linear, Quadratic Polynomials etc. Value of a polynomial for a given value of the variable Zeroes of a polynomial Remainder theorem Factor theorem Factorisation of a quadratic polynomial by splitting the middle term Factorisation of algebraic epressions by using the Factor theorem Algebraic identities ( + y) = + y + y ( y) = y + y y = ( + y) ( y) ( + a) ( + b) = + (a + b) + ab ( + y + z) = + y + z + y + yz + z ( + y) = + y + y + y = + y + y ( + y) ( y) = y + y y = y y ( y) 90504
4 EXEMPLAR PROBLEMS + y = ( + y) ( y + y ) y = ( y) ( + y + y ) + y + z yz = ( + y + z) ( + y + z y yz z) (B) Multiple Choice Questions Sample Question : If + k + 6 = ( + ) ( + ) for all, then the value of k is (A) (B) (C) 5 (D) Solution : Answer (C) EXERCISE. Write the correct answer in each of the following :. Which one of the following is a polynomial? (A) (B) (C) + (D). is a polynomial of degree + (A) (B) 0 (C) (D). Degree of the polynomial 4 4 + 0 + 0 5 + 5 + 7 is (A) 4 (B) 5 (C) (D) 7 4. Degree of the zero polynomial is (A) 0 (B) (C) Any natural number (D) 5. If ( ) Not defined = +, then ( ) p p is equal to (A) 0 (B) (C) 4 (D) 8 + 6. The value of the polynomial 5 4 +, when = is (A) 6 (B) 6 (C) (D) 90504
POLYNOMIALS 5 7. If p() = +, then p() + p( ) is equal to (A) (B) (C) 0 (D) 6 8. Zero of the zero polynomial is (A) 0 (B) (C) Any real number (D) Not defined 9. Zero of the polynomial p() = + 5 is (A) 5 (B) 5 (C) 0. One of the zeroes of the polynomial + 7 4 is (A) (B) (C). If 5 + 5 is divided by +, the remainder is 5 (D) (D) (A) 0 (B) (C) 49 (D) 50. If + is a factor of the polynomial + k, then the value of k is (A) (B) 4 (C) (D). + is a factor of the polynomial (A) + + (B) + + + (C) 4 + + + (D) 4 + + + + 4. One of the factors of (5 ) + ( + 5) is (A) 5 + (B) 5 (C) 5 (D) 0 5. The value of 49 48 is (A) (B) 477 (C) 487 (D) 497 6. The factorisation of 4 + 8 + is (A) ( + ) ( + ) (B) ( + ) ( + ) (C) ( + ) ( + 5) (D) ( ) ( ) 7. Which of the following is a factor of ( + y) ( + y )? (A) + y + y (B) + y y (C) y (D) y 8. The coefficient of in the epansion of ( + ) is (A) (B) 9 (C) 8 (D) 7 y 9. If y + = (, y 0 ), the value of y is 5 90504
6 EXEMPLAR PROBLEMS (A) (B) (C) 0 (D) 0. If 49 b = 7 + 7, then the value of b is (A) 0 (B) (C). If a + b + c = 0, then a + b + c is equal to (A) 0 (B) abc (C) abc (D) abc (C) Short Answer Questions with Reasoning Sample Question : Write whether the following statements are True or False. Justify your answer. Solution : 5 + is a polynomial (ii) 4 6 + (D) is a polynomial, 0 False, because the eponent of the variable is not a whole number. (ii) True, because 6 + = 6 +, which is a polynomial. EXERCISE.. Which of the following epressions are polynomials? Justify your answer: 8 (ii) (iii) 5 (iv) 5 + 5 + 7 (v) ( )( 4) (vi) + (vii) a a + 4 a 7 (viii) 7 90504
POLYNOMIALS 7. Write whether the following statements are True or False. Justify your answer. (ii) A binomial can have atmost two terms Every polynomial is a binomial (iii) A binomial may have degree 5 (iv) Zero of a polynomial is always 0 (v) A polynomial cannot have more than one zero (vi) The degree of the sum of two polynomials each of degree 5 is always 5. (D) Short Answer Questions Sample Question : Check whether p() is a multiple of g() or not, where (ii) Solution : p() = +, g() = Check whether g() is a factor of p() or not, where p() = 8 6 4 +, g() = 4 p() will be a multiple of g() if g() divides p(). Now, g() = = 0 gives = Remainder = p = + = 8 + = 7 7 7 Since remainder 0, so, p() is not a multiple of g(). (ii) g() = = 0 gives = 4 4 g() will be a factor of p() if p = 0 (Factor theorem) 4 Now, p = 8 6 4 + 4 4 4 4 90504
8 EXEMPLAR PROBLEMS Since, 7 9 = 8 6 + = 0 64 6 p = 0, so, g() is a factor of p(). 4 Sample Question : Find the value of a, if a is a factor of a + + a. Solution : Let p() = a + + a Since a is a factor of p(), so p(a) = 0. i.e., a a(a) + a + a = 0 a a + a + a = 0 a = Therefore, a = Sample Question : Without actually calculating the cubes, find the value of 48 0 8. (ii)without finding the cubes, factorise ( y) + (y z) + (z ). Solution : We know that + y + z yz = ( + y + z) ( + y + z y yz z). If + y + z = 0, then + y + z yz = 0 or + y + z = yz. We have to find the value of 48 0 8 = 48 + ( 0) + ( 8). Here, 48 + ( 0) + ( 8) = 0 So, 48 + ( 0) + ( 8) = 48 ( 0) ( 8) = 77760 (ii) Here, ( y) + (y z) + (z ) = 0 Therefore, ( y) + (y z) + (z ) = ( y) (y z) (z ). EXERCISE.. Classify the following polynomials as polynomials in one variable, two variables etc. + + (ii) y 5y (iii) y + yz + z (iv) y + y + 90504
POLYNOMIALS 9. Determine the degree of each of the following polynomials : (ii) 0 (iii) 9 + 5 (iv) y ( y 4 ). For the polynomial + + 7 6, write 5 the degree of the polynomial (ii) the coefficient of (iii) the coefficient of 6 (iv) the constant term 4. Write the coefficient of in each of the following : π 6 + (ii) 5 (iii) ( ) ( 4) (iv) ( 5) ( + ) 5. Classify the following as a constant, linear, quadratic and cubic polynomials : + (ii) (iii) 5 t 7 (iv) 4 5y (v) (vi) + (vii) y y (viii) + + (i) t () 6. Give an eample of a polynomial, which is : monomial of degree (ii) binomial of degree 0 (iii) trinomial of degree 7. Find the value of the polynomial 4 + 7 5, when = and also when =. 8. If p() = 4 +, evaluate : p() p( ) + p 9. Find p(0), p(), p( ) for the following polynomials : p() = 0 4 (ii) p(y) = (y + ) (y ) 0. Verify whether the following are True or False : is a zero of 90504
0 EXEMPLAR PROBLEMS (ii) is a zero of + (iii) (iv) 4 5 is a zero of 4 5y 0 and are the zeroes of t t (v) is a zero of y + y 6. Find the zeroes of the polynomial in each of the following : p() = 4 (ii) g() = 6 (iii) q() = 7 (iv) h(y) = y. Find the zeroes of the polynomial : p() = ( ) ( + ). By actual division, find the quotient and the remainder when the first polynomial is divided by the second polynomial : 4 + ; 4. By Remainder Theorem find the remainder, when p() is divided by g(), where p() = 4, g() = + (ii) p() = + 4 + 50, g() = (iii) p() = 4 + 4, g() = (iv) p() = 6 + 4, g() = 5. Check whether p() is a multiple of g() or not : p() = 5 + 4, g() = (ii) p() = 4 + 5, g() = + 6. Show that : + is a factor of 69 + +. (ii) is a factor of + 9 +. 7. Determine which of the following polynomials has a factor : + 6 4 (ii) 4 + 8. Show that p is a factor of p 0 and also of p. 9. For what value of m is m + 6 divisible by +? 0. If + a is a factor of 5 4a + + a +, find a.. Find the value of m so that be a factor of 8 4 + 4 6 + 0 + m. 90504
POLYNOMIALS. If + is a factor of a + + 4a 9, find the value of a.. Factorise : + 9 + 8 (ii) 6 + 7 (iii) 7 5 (iv) 84 r r 4. Factorise : 7 + 0 (ii) 6 + 6 (iii) + 4 4 (iv) + 5. Using suitable identity, evaluate the following: 0 (ii) 0 0 (iii) 999 6. Factorise the following: 4 + 0 + 5 (ii) 9y 66yz + z (iii) + 7. Factorise the following : 9 + (ii) 9 + 4 8. Epand the following : (4a b + c) (ii) (a 5b c) (iii) ( + y z) 9. Factorise the following : 9 + 4y + 6z + y 6yz 4z (ii) 5 + 6y + 4z 40y + 6yz 0z (iii) 6 + 4y + 9z 6y yz + 4 z 0. If a + b + c = 9 and ab + bc + ca = 6, find a + b + c.. Epand the following : y (a b) (ii) +. Factorise the following : 64a a + 48a (iii) 4 90504
EXEMPLAR PROBLEMS (ii) 8 6 + + + 5 5 5 p p p. Find the following products : 4. Factorise : + y y + 4y 4 (ii) ( ) ( 4 + + ) + 64 (ii) a b 5. Find the following product : ( y + z) (4 + y + 9z + y + yz 6z) 6. Factorise : a 8b 64c 4abc (ii) a + 8b 7c + 8 abc. 7. Without actually calculating the cubes, find the value of : 5 + 6 8. Without finding the cubes, factorise ( y) + (y z) + (z ) 9. Find the value of + y y + 64, when + y = 4 (ii) 8y 6y 6, when = y + 6 (ii) (0.) (0.) + (0.) 40. Give possible epressions for the length and breadth of the rectangle whose area is given by 4a + 4a. (E) Long Answer Questions Sample Question : If + y = and y = 7, find the value of + y. Solution : + y = ( + y) ( y + y ) = ( + y) [( + y) y] = [ 7] = 6 = 756 90504
POLYNOMIALS Alternative Solution : + y = ( + y) y ( + y) = 7 = [ 7] = 6 = 756 EXERCISE.4. If the polynomials az + 4z + z 4 and z 4z + a leave the same remainder when divided by z, find the value of a.. The polynomial p() = 4 + a + a 7 when divided by + leaves the remainder 9. Find the values of a. Also find the remainder when p() is divided by +.. If both and are factors of p + 5 + r, show that p = r. 4. Without actual division, prove that 4 5 + + is divisible by +. [Hint: Factorise + ] 5. Simplify ( 5y) ( + 5y). 6. Multiply + 4y + z + y + z yz by ( z + y). 7. If a, b, c are all non-zero and a + b + c = 0, prove that a b c + + =. bc ca ab 8. If a + b + c = 5 and ab + bc + ca = 0, then prove that a + b + c abc = 5. 9. Prove that (a + b + c) a b c = (a + b ) (b + c) (c + a). 90504