UNIT- POLYNOMIALS Downloded From: www.jsuniltutoril.weebly.com [Yer] UNIT- POLYNOMIALS 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) = x 4 + 0x 4 + 5x + 5x + K nd g(x) = x + 7 Since P(x) exctly divisible by g(x) r (x) = 0 (Ans : K= - 9) now x + 7 3 4 3 4 3 x 3 x 4 x 3 x 0 x 5 x 5 x K x 7 x ------------- 3x 3 + 5 x 3x 3 + x ------------------- 4x + 5 x 4x + 8x ------------------ -3x + K - 3x - 9 ---------------- K + 9 ------------ K + 9 = 0 www.jsuniltutoril.weebly.com Pge 6
UNIT- POLYNOMIALS Downloded From: www.jsuniltutoril.weebly.com [Yer] K= -9. If two zeros of the polynomil f(x) = 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 x 35 x 4x + 4 3 x 6 x 6 x 38 x 35 4 3 x 4 x x ----------------- -x 3 7x + 38x - x 3 + 8x x ----------------------- -35x + 40x 35-35x + 40x 35 ------------------------ 0 www.jsuniltutoril.weebly.com Pge 7
UNIT- POLYNOMIALS Downloded From: www.jsuniltutoril.weebly.com [Yer] ------------------------ 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) = x 4 x + 5 + = b 4 = = c 5 + = ( + ) Substitute then we get, ( - ) = ( + ) - 4 + = - www.jsuniltutoril.weebly.com Pge 8
UNIT- POLYNOMIALS Downloded From: www.jsuniltutoril.weebly.com [Yer] 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) = x + 8 x + 6 + = -8 nd = 6 (Ans: x + 4 x 3 +, x 3 - x 6 3 3 + ) 3 ) Let two zeros re nd Sum = + =. = 8 6 = 4 3 Product = x =. 6 Required Q.P is x + 4 x 3 6 b) Let two Zeros re + nd + sum = + ++ = + + = + www.jsuniltutoril.weebly.com Pge 9
UNIT- POLYNOMIALS Downloded From: www.jsuniltutoril.weebly.com [Yer] = + ( ) fter solving this problem, 3 We get = 3 Product = ( + )(+ ) = + + + = + Substitute this sum, 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(x) the quotient is x - 3x 5 nd the reminder is -5x + 8.Find the polynomil g(x). (Ans:4 x +7x+) Ans: p(x) = g (x) q (x) + r (x) g(x) = p ( x) r ( x) q ( x) let p(x) = 4x 4 5x 3 39x 46x q(x) = x 3x 5 nd r (x) = -5x + 8 www.jsuniltutoril.weebly.com Pge 0
UNIT- POLYNOMIALS Downloded From: www.jsuniltutoril.weebly.com [Yer] now p(x) r(x) = 4x 4 5x 3 39x 4x - 0 when p ( x) r ( x) q ( x) = 4x + 7x + g(x) = 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) = 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 www.jsuniltutoril.weebly.com Pge
UNIT- POLYNOMIALS Downloded From: www.jsuniltutoril.weebly.com [Yer] 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 4 4 (Ans:-,,-6,,-7) 5 5 Ans: Let p(x) = x 4x +5 + = b = 4 = = c = 5 ) + = ( + ) - Substitute to get = + = - b) + = substitute, then we get = + = 5 4 b) ( - ) = ( + ) - 4 Therefore we get, ( - ) = - 6 www.jsuniltutoril.weebly.com Pge
UNIT- POLYNOMIALS Downloded From: www.jsuniltutoril.weebly.com [Yer] d) + = = 5 4 + = 5 e) 3 + 3 = ( + )( + - ) Substitute this, to get, 3 + 3 = -7 0. Obtin ll the zeros of the polynomil p(x) = 3x 4 5x 3 + 7x +5x 6 if two zeroes re / 3 nd / 3. (Ans:3,). Give exmples of polynomils p(x), g(x), q(x) nd r(x) which stisfy the division lgorithm.. deg p(x) = deg q(x) b. deg q(x) = deg r(x) c. deg q(x) = 0.. If the rtios of the polynomil x 3 +3bx +3cx+d re in AP, Prove tht b 3-3bc+ d=0 Ans: Let p(x) = x 3 + 3bx + 3cx + d nd,, r re their three Zeros but zero re in AP let = m n, = m, r = m + n sum = + + r = b substitute this sum, to get = m= b www.jsuniltutoril.weebly.com Pge 3
UNIT- POLYNOMIALS Downloded From: www.jsuniltutoril.weebly.com [Yer] 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 )m = d b [( ) 3b 3c b ( ) ] ( ) = d Simplifying we get b 3 3bc + d = 0 3. Find the number of zeros of the polynomil from the grph given. (Ans:) www.jsuniltutoril.weebly.com Pge 4
UNIT- POLYNOMIALS Downloded From: www.jsuniltutoril.weebly.com [Yer] 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 k = n + n m q 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 m q 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.n=0 (Ans: =/,c=5) 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)= x 4 + x 3 x + x so tht the resulting polynomil is exctly divisible by x +x-3. (Ans: x-) www.jsuniltutoril.weebly.com Pge 5