POLYNOMIALS UNIT- It is not once nor twice but times without number tht the sme ides mke their ppernce in the world.. Find the vlue for K for which x 4 + 0x 3 + 5x + 5x + K exctly divisible by x + 7. Ans: Let P( x 4 + 0x 4 + 5x + 5x + K nd g( x + 7 Since P( exctly divisible by g( r ( 0 now x + 7 3 x + 3x + 4x 3 4 3 x + 0x + 5x + 5x + K K + 9 0 4 3 x + 7x ------------- 3x 3 + 5 x 3x 3 + x ------------------- 4x + 5 x 4x + 8x ------------------ -3x + K - 3x - 9 ---------------- K + 9 ------------ (Ans : K - 9) K -9. If two zeros of the polynomil f( x 4-6x 3-6x + 38x 35 re ± 3.Find the other zeros. (Ans:7, -5) Ans: Let the two zeros re + 3 nd - 3 Sum of Zeros + 3 + - 3 4 Product of Zeros ( + 3 )( - 3 ) 4 3 Qudrtic polynomil is x (sum) x + Product
x 4x + x x 35 4 3 x x x x 6 6 + 38 35 4 3 x 4x + x ----------------- -x 3 7x + 38x - x 3 + 8x x ----------------------- -35x + 40x 35-35x + 40x 35 ------------------------ 0 ------------------------ x x 35 0 (x 7)(x + 5) 0 x 7, -5 other two Zeros re 7 nd -5 3. Find the Qudrtic polynomil whose sum nd product of zeros re +, Ans: sum Product Q.P X (sum) x + Product +. x ( ) x + 4. If, re the zeros of the polynomil x 4x + 5 find the vlue of ) + b) ( - ). (Ans: ) -, b) 6) Ans: p ( x 4 x + 5 b 4 + c 5 + ( + ) Substitute then we get, + - ( - ) ( + ) - 4 Substitute, we get ( - ) - 6
5. If, re the zeros of the polynomil x + 8x + 6 frme Qudrtic polynomil whose zeros re ) nd b) +, +. Ans: p ( x + 8 x + 6 + -8 nd 6 (Ans: x + 4 x +, x 3-3 x 3 + ) 6 3 3 ) Let two zeros re nd Sum +. + 8 4 6 3 Product x. 6 Required Q.P is x + 4 x + 3 6 b) Let two Zeros re + nd + sum + ++ + + + + ( + ) + fter solving this problem, 3 We get 3 Product ( + )(+ ) + + + + + Substitute this sum, 3
3 We get 3 Required Q.P. is x 3 3 - x + 3 3 6. On dividing the polynomil 4x 4-5x 3-39x - 46x by the polynomil g( the quotient is x - 3x 5 nd the reminder is -5x + 8.Find the polynomil g(. (Ans:4 x +7x+) Ans: p( g ( q ( + r ( p( r( g( q( let p( 4x 4 5x 3 39x 46x q( x 3x 5 nd r ( -5x + 8 now p( r( 4x 4 5x 3 39x 4x - 0 p( r( when 4x + 7x + q( g( 4x + 7x + 7. If the squred difference of the zeros of the qudrtic polynomil x + px + 45 is equl to 44, find the vlue of p. (Ans:± 8). Ans: Let two zeros re nd where > According given condition ( - ) 44 Let p( x + px + 45 b + - p p c 45 45 now ( - ) 44 ( + ) 4 44 (-p) 4 (45) 44 Solving this we get p ± 8 8. If, re the zeros of Qudrtic polynomil such tht + 4, - 8. Find Qudrtic polynomil hving nd s its zeros. (Ans: k(x 4x + 8)) Ans: + 4-8 ----------- 3 4
3 6, 6 Work the sme wy to + 4 So, 8 Q.P is x (sum) x + product x (6+8) x + 6 x 8 Solve this, it is k (x 4x + 8) 9. If & ß re the zeroes of the polynomil x 4x + 5, then find the vlue of. + ß b. / + / ß c. ( ß) d. / + /ß e. 3 + ß 3 Ans: Let p( x 4x +5 b 4 + c 5 4 4 (Ans:-,,-6,,-7) 5 5 ) + (+) - Substitute to get + - + b) + substitute, then we get + 5 4 b) (-) (+) - 4 Therefore we get, (-) - 6 d) + + + 4 5 5 e) 3 + 3 (+)( + - ) Substitute this, to get, 3 + 3-7 5
0. Obtin ll the zeros of the polynomil p( 3x 4 5x 3 + 7x +5x 6 if two zeroes re / 3 nd / 3. (Ans:3,). Give exmples of polynomils p(, g(, q( nd r( which stisfy the division lgorithm.. deg p( deg q( b. deg q( deg r( c. deg q( 0.. If the rtios of the polynomil x 3 +3bx +3cx+d re in AP, Prove tht b 3-3bc+ d0 Ans: Let p( x 3 + 3bx + 3cx + d nd,, r re their three Zeros but zero re in AP let m n, m, r m + n b sum ++ r substitute this sum, to get m b Now tking two zeros s sum + r +r c 3c (m-n)m + m(m+n) + (m + n)(m n) Solve this problem, then we get 3b 3c n Product r d (m-n)m (m+n) d (m n d )m b [( ) 3b 3c ( ) Simplifying we get b 3 3bc + d 0 b ] ( ) d 6
3. Find the number of zeros of the polynomil from the grph given. (Ans:) 4. If one zero of the polynomil 3x - 8x +k+ is seven times the other, find the zeros nd the vlue of k (Ans k /3) Self Prctice 4. If (n-k) is fctor of the polynomils x +px+q & x + m x+n. Prove tht n q k n + m p Ans : since (n k) is fctor of x + px + q (n k) + p(n- k) + q 0 And (n k) + m(n k) + n 0 Solve this problem by yourself, k n + n q m p SELF PRACTICE 6. If, ½ re the zeros of px +5x+r, prove tht p r. 7. If m, n re zeroes of x -5x+c, find the vlue of nd c if m + n m.n0 (Ans: /,c5) 8. Wht must be subtrcted from 8x 4 + 4x 3 x + 7x 8 so tht the resulting polynomil is exctly divisible by 4x +3x-. (Ans: 4x 0) 9. Wht must be dded to the polynomil p( x 4 + x 3 x + x so tht the resulting polynomil is exctly divisible by x +x-3. (Ans: x-) 7