Solved problems strictly according to the syllabus.
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1. 2. Polynomials Class IX Determine which of the following polynomials has (x + 1) a factor: (iii) x4 + 3x3 + 3x2 + x + I (iv) x3 - x2 - (2 + 42)x + 42 Answer (i) If (x + 1) is a factor of p(x) = x3 + x2 + x + 1, p(-l) must be zero. Here, p(x) = x3 + x2 + x + 1 - (-1)3 + (-1)2 + + 1 = -1 +1-1+1=0 Therefore, x + 1 is a factor of this polynomial (ii) If (x + 1) is a factor of p(x) = x4 + x3 + x2 + x + 1, p(-l) must be zero. Here, p(x) = x4 + x3 + x2 + x + 1 - (-1)4 + (-1)3 + (-1)2 + + 1 Therefore, x + 1 is not a factor of this polynomial (iii)lf (x + 1) is a factor of polynomial p(x) = x 4 + 3x3 + 3x2 + x + 1, p(- 1) must be 0. Therefore, x + 1 is not a factor of this polynomial. (iv) If (x + 1) is a factor of polynomial p(x) = x3 - x2 - (2 + 42)x + 42, p(- 1) must be 0. ( - (-1)2- (2 + 42) + 42 = -l - I +2 + -02 + 42 242 Therefore„ x + 1 is not a factor of this polynomial. Use the Factor Theorem to determine whether g(x) is a factor of p(x) in each of the following cases: (i)p(x) = 2x3 + x2 - 2x - 1, g(x) = x + 1 (ii) p(x) = x3 + 3x2 + 3x + 1, g(x) = x + 2 (iii) p(x) = x3 - 4 x2 + x + 6, g(x) = x - 3 Answer (i) If g(x) = x + 1 is a factor of given polynomial p(x), p(- 1) must be zero. p(x) = 2x3 + x2 - 2x - 1 1) = + (-1)2 - 2(-1) - 1
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3. Hence, g(x) = x + 1 is a factor of given polynomial. (ii) If g(x) = x + 2 is a factor of given polynomial p(x), p(- 2) must be 0. = X3 +3X2 + 3X + 1 p(-2) = (-2)3 + + 3(-2) + 1 -8 + 12 -6+1 --1 As, p(-2) 0 Hence g(x) = x + 2 is not a factor of given polynomial. (iii) If g(x) = x - 3 is a factor of given polynomial p(x), p(3) must be 0. (3)3 - +3 +6 = 27-36+9=0 Therefore„ g(x) = x - 3 is a factor of given polynomial. Find the value of k, if x - 1 is a factor of p(x) in each of the following cases:(i) p(x) = (ii)p(x) = 2x2 + kx + 42 (iii) p(x) = kx2 - m12x + 1 (iv) p(x) = kx2 - 3x + k Answer (i) If x - 1 is a factor of polynomial p(x) = Therefore, value of k is -2. x2 + x + k, then 2X2 + kX + 42, then (ii) If x - 1 is a factor of polynomial p(x) = k = -2 - 42 = -(2 + 42) Therefore, value of k is -(2 + 42). kx2 - A12x + 1, then (iii) If x - 1 is a factor of polynomial p(x) = Therefore, value of k is 42 - 1. 2 - 3x + k, then (iv) If x - 1 is a factor of polynomial p(x) = kx
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4. 5. Therefore, value of k is 3/2. Factorise: (i) 12x2 + 7.r + I (ii) 2x2 + 7x + 3 (iii) 6x2 + 5x - 6 (iv) 3x2 -x - 4 Answer (i) 12x2 +7x+ 1 = 12x2 - 4x - 3x+ 1 (ii) 2x2 + + 3 = 2X2 + 6X + X + 3 (iii) 6x2 + 5x - 6 = 6x2 + 9x - 4x - 6 (iv) 3x2 -x - 4 = 3x2 — 4 x + 3 x — 4 Factorise: (i) x3 - 2x2 -x + 2 (ii) x3 - 3x2 - 9x - 5 (iii) x3 + 13x2 + 32x + 20 (iv) 2y3 + Y2 - 2y - 1 Answer (i) Let p(x) = x3 - 2x2 - x + 2 Factors of 2 are ±1 and ± 2 By trial method, we find that So, (x+l ) is factor of p(x) Now, p(x) = x3 - 2x2 - x + 2 p(-l) = (-1)3 - Therefore, (x+l) is the factor of p(x)
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X3 — 2X2 —X + 2 2 --3x2 —x + 2 — 3x2 — 3x 2x+2 Now, Dividend = Divisor x Quotient + Remainder = (x+l) {x(x-l) -2(x-1)} (ii) Let p(x) = x3 - 3x2 - 9x - 5 Factors of 5 are ±1 and ±5 By trial method, we find that So, (x-5) is factor of p(x) Now, p(x) = x3 - 2x2 - x + 2 p(5) = (5)3 - - 9(5) - 5 = 125 - 75 Therefore, (x-5) is the factor of p(x) - 45 x-S x2 + 2.1 I x 3 3x2 — 9x x 3 5x 2 2x2 9x —5 —5 2x2 IOx Now, Dividend = Divisor x Quotient + Remainder = (X-5) {X(X+I) 4-1 (X + I ) }
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(iii) Let p(x) = + 13x2 + 32x + 20 Factors Of 20 are ± l, and By trial method, we find that So, (x+l ) is factor of p(x) Now, = X3 + 13X2 + 32X + 20 - 3 + + 32(-1) + 20 — —-1 13 - Therefore, (x+l) is the factor of p(x) X2 12X 20 x3+ 13x2 + 20 2 12x2 + 32x + 20 12x2 + 12x 20x + 20 20x + 20 0 32 + 20 = o Now, Dividend = Divisor x Quotient + Remainder + 12X + 20) = (x +1) (x2 2x + + 20) = (X-5) {X(X+2) = (x-5) (x+2) (x+10) (iv) Let p(y) = 2y3 + Y2 - 2y - 1 Factors of ab = 2x (-1) = -2 are ±1 and ±2 By trial method, we find that So, (y-1 ) is factor of p(y) Now, p(y) = 2y3 + Y2 - 2y - 1 p(l) + - = 2+1 2- Therefore, (y-1) is the factor of p(y)
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y-1 2y2 + 3y + 1 2y3 + Y2 — 2y 2y3 —2y2 3y2 — 2y — 3y —1 —1 Now, Dividend = Divisor x Quotient + Remainder = (y-I) 4-1) (y +1) Use suitable identities to find the following products: 6. (i) (x + 4) (x + 10) (iv) + 3/2) (Y2 - 3/2) Answer (ii) (x + 8) (x — 10) (iii) (3x + 4) (3x — 5) (i) Using identity, (x + a) (x + b) = x2 + (a + b) x + ab In (x +4) (x+ 10), and 10 Now, (X + 4) (X + 10) X2 + (4 + + (4 X 10) = X2 + 40 (ii) (x+ 8) (x- 10) Using identity, (x + a) (x + b) = x2 + (a + b) x + ab Here, a = 8 and b = -10 (X + 8) (X — 10) = X2 + {8 4-(— 10) } X + 10)) = x2 + (8 — 10)x — 80 2x 80 (iii) (3x + 4) (3x 5) Using identity, (x + a) (x + b) = x2 + (a + b) x + ab Here, x = 3x, and b = -5 = + 3x(4 - 5) - 20 = 9x2 - 3x - 20 (iv) + 3/2) (Y2 - 3/2) Using identity, (x + y) (x -y) = x2 2
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, and y = 3/2 Here x = (Y2 + 3/2) (Y2 - 3/2) = (y2)2 - (3/2)2 Using identity, (x + y) (x -y) = Here, x = 3 and y = 2x (3 - 2x) (3 + 2x) = 32 - (2x)2 9 - 4x2 2 Evaluate the following products without multiplying directly: 7. (i) 103 x 107 (ii) 95 x 96 (iii) 104 x 96 Answer (i) 103 x 107 = (100 + 3) (100 + 7) Using identity, (x + a) (x + b) = x2 + (a + b) x + ab Here, x = 100, a = 3 and b = 7 103 x 107 = (100 + 3) (100 + 7) = (100)2 + (3 + 7)10 + (3 x 7) = 10000 + 100 + 21 = 11021 (ii) 95 x 96 = (90 + 5) (90 + 4) Using identity, (x + a) (x + b) = x2 + (a + b) x + ab Here, x = 90, a = 5 and b = 4 95 x 96 = (90 + 5) (90 + 4) = 902 + + 6) + (5 x 6) = 8100 + (11 x 90) + 30 = 8100 + 990 + 30 = 9120 (iii) 104 x 96 = (100 + 4) (100 - 4) Using identity, (x + y) (x -y) = x2 - y 2 Here, x = 100 and y = 4 104 x 96 = (100 + 4) (100 - 4) = (100)2 - (4)2 = 10000 - 16 = 9984 8. Factorise the following using appropriate identities: (i) 9x2 + 6xy + (ii) 4y2 - 4y + 1 (iii) x2 - y2/100 Answer (i) 9x2 + 6xy + Y2 = (3x) 2 + (2>
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9. Using identity, a2 - b2 = (a + b) (a - b) Here, a = x and b = (y/ 10) x2- y2/100 = x2- (y/ 10) — 2- (x- y/10) (x+ y/10) Expand each of the following, using suitable identities: (iv) (3a — 7b — c)2 Answer (i) (x + + 4z)2 (ii) (2x —y + z)2 (v) (—2x + 5y — 3z)2 (iii) (—2x + 3y + 2z)2 (vi) [1/4 a - 1/2 b + 112 Using identity, (a + b + = a2 + b2 + c2 + 2ab + 2bc + 2ca Here, a = x, b = 2y and c = 4z (x + 2y + + (2y)2 + (4z)2 + (2>
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Answer (i) 4x2 + 9y2 + 16z2 + 12xy - 24yz - 16xz Using identity, (a + b + = a2 + b2 + c2 + 2ab + 2bc + 2ca 4x2 + 9y2 + 16z2 + 12xy - 24yz - 16xz = (2x)2 + (3y)2 + (-4z)2 + + + = (2x + 3 Y' - 4z)2 = (2x + 3y - 4z) (2x + 3y - 4z) (ii) 2x2 + Y2 + 8z2 - 242 xy + 442 yz - 8xz Using identity, (a + b + = a2 + b2 + c2 + 2ab + 2bc + 2ca 2x2 + Y2 + 8z2 - 2€2 xy + 4Q2yz - 8xz = ( -€21) 2 + (y) 2 (2N2z)2 + = (-Q2x y 2N2z)2 = (-€2x y 2N2z) (-€2x y 2N/2z) Write the following cubes in expanded form: 11. (i) (2x + 1)3 (ii) (2a — 3b)3 (iii) [3/2 x + 113 Answer Using identity, (a + = a3 + b3 + 3ab(a + b) 3 = (2x)3 + 13 + + 1) = + I + 6x(2x + ) = 8x3 + 12x2 + 6x + (ii) (2a - 3b)3 Using identity, (a - = a3 - b3 - 3ab(a - b) (2a 3b/ = (2a)3 - (3b/ - - 3b) = 8a3 - 27b3 - 18ab(2a - 3b) = 8a3 - 27b3 - 36a2b + 54ab2 (iii) [3/2 x + Using identity, (a + = a3 + b3 + 3ab(a + b) [3/2 x + 113 = (3/2 x/ + 13 + + 1) = 27/8x3+1+9/2x(3/2x+1) = 27/8x3+1 +27/4x2+9/2x = 27/8++27/4x2+9/2x+1 (iv) [x - 2/3 yr Using identity, (a - = a3 - b3 - 3ab(a - b) 3 = (x/ - (2/3 y/ - - 2/3 y) [x - 2/3 y] = x3 - 8/27y3 - 2xy(x - 2/3 y) = x3 - 8/27y3 - 2x2y + 4/3xy2 12. Evaluate the following using suitable identities: (i) (99)3 (ii) (102)3 (iii) (998)3 Answer (iv) [x - 2/3 y13
Factorise : 27x3 + y3 + z3 - 9xyz 16. Answer 27x3 + y3 + z3 - 9xyz = (3x)3 + y3 + z3 - 3x3xyz Using identity, x3 + y3 + z3 - 3xyz = (x + y + + Y2 + z2 27x3 + Y'3 + _ 9xyz = (3x + y + z) {(3x)2 + Y2 + z2 - 3xy - yz - 3xz} = + y + z) (9x2 + + - yz - 3xz) - xy - yz - xz) Verify that: x3 + y3 + z3 - 3xyz = 1/2(x + y + z) [(x - + (y - + (z - x)2] 17. Answer We know that, x3 + y3 + z3 - 3xyz = (x + y + + Y2 + z2 - xy - yz - xz) + y3 + _ 3 xyz + y + z) + + z2 — XY — Y z — R,) = 1/2(x + y + z) (2x2 + 2y2 + 2z2 - 2xy - 2yz - 2xz) = 1/2(x + y + z) [(x2 + Y2 -2xy) + (Y2 + z2 - 2yz) + (x2 + z2 - 2xz)] 18. If x + y + z = 0, show thatx3 +y3 + z3 = 3xyz. Answer We know that, x3 + y3 + z3 - 3xyz = (x + y + + Y2 + z2 - xy - yz - xz) Now put (x + y + z) = 0, x3 + y3 + z3 - 3xyz = + Y2 + z2 - xy - yz - xz) 19. Without actually calculating the cubes, find the value of each of the following: (i) (-12)3 + (7)3 + (5)3 (ii) (28)3 + (-15)3 + (-13)3 Answer (i) (-12) 3 + (7)3 + (5)3 and z = 5 We observed that, x + y + z = -12 + 7 + 5 = 0 We know that if, x + y + z = 0, then x3 + y3 + z3 = 3xyz - -1260 (ii) (28)3 + (-15)3 + (-13)3 Let x -28, y = -15 and z = -13 We observed that, x + y + z = 28 - 15 - 13 = 0
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We know that if, x + y + z = 0, then x3 + y3 + z3 = 3xyz (28/ + (-15/ + (-13/ = = 16380 20. Give possible expressions for the length and breadth of each of the following rectangles, in which their areas are given: (i) Area : 25a2 - 35a + 12 (ii) Area : 35 Y2 + 13y - 12 Answer (i) Area : 25a2 - 35a + 12 Since, area is product of length and breadth therefore by factorizing the given area, we can know the length and breadth of rectangle. 25a2 - 35a + n = 25a2 - 15a -20a + n = 5a(5a - 3) - 4(5a - 3) Possible expression for length = 5a - 4 Possible expression for breadth = 5a - 3 (ii) Area : 35 + 13y - 12 35 + 13y- n = 35y2 - 15y + 28y - 12 = 5y(7y - 3) + 4(7y - 3) Possible expression for length = (5y + 4) Possible expression for breadth = (7y - 3) Page No: 50 21. What are the possible expressions for the dimensions of the cuboids whose volumes are given below? (i) Volume : 3x2 - 12x (ii) Volume : 12ky2 + 8ky - 20k Answer (i) Volume : 3x2 - 12x Since, volume is product of length, breadth and height therefore by factorizing the given volume, we can know the length, breadth and height of the cuboid. 3x2 - 12x = 3x(x - 4) Possible expression for length Possible expression for breadth = x Possible expression for height = (x - 4)
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(ii) Volume : 1202 + 8ky - 20k Since, volume is product of length, breadth and height therefore by factorizing the given volume, we can know the length, breadth and height of the cuboid. 12/02 + 8ky - 20k = 4k(3y2 + 2y - 5) = 4k(3y2 +5y - 3y - 5) = +5) - I(3y + 5)] Possible expression for length Possible expression for breadth Possible expression for height
Discussion
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