Higher Algebra

ຑຈຶ ຆະ຃ະຌຈິ ຆຌໄ ສູຄ 1- 6t + 2t2 + t3 1  t  2t 2 1  5t  t 2 - (1- t - )2t2 0 - 5t + 4i2 + t3 ( )- - 5t + 5t2 +10t3 0 - t 2 - 9t3 ( )- - t2 + t3 + 2t4 0 - 10t3 - 2t4 ກຳຌຫຳຌສຌິໄ ສຸຈລຄ຺ ຽມໃ ຬຼື ກຳຌຫຳຌມຂີ ຌໄ ຂຬຄຏຌ຺ ແຈອໄ ຍ Q ຽ຋ໃ ຳ຺ ກຍກຳຌກຳຌຈ຺ ຂຬຄຑຈ຺ ຽລກ ຈໃ ຄຌຌໄ ຏຌ຺ ຫຳຌຂຳໄ ຄຽ຋ຄິ ຾ມໃ ຌ ( ) ( )( ) ( )1- 6t + 2t2 + t3 = 1- t - 2t2 1- 5t - t2 - 10t3 + 2tt4 ຉວ຺ ຢໃ ຳຄ 2: ຫຳຌຑະຫຸຑຈ຺ A = 9x - 7x2 - x3 + x4 ຈວໄ ງ B = 1+ x + x2 ຉຳມກຳລຄຂຌຶໄ ອຬຈຂຌໄ 3 ( )( )ຽຫຌວໃ ຳ 9x - 7x2 - x3 + x4 = 1+ x - x2 8 - 17x +18x2 - 36x3 +(55 - 36x) x4 ຉວ຺ ຢໃ ຳຄ 3: ຫຳຌ A = 1 ຈວໄ ງ B = 1  x ຉຳມກຳລຄຂຌຶໄ ອຬຈຂຌໄ n ຂຌໄ ຉຬຌ຋ຳຬຈິ ຫຳຌອຬຈຂຌໄ 2 ກໃ ຬຌ. 1 = (1- x)1+ x ອຬຈຂຌໄ 0 1 = (1- x)(1+ x)+ x2 ອຬຈຂຌໄ 1 ( )1 = (1- x) 1+ x + x2 + x3 ອຬຈຂຌໄ 2  ( )1= (1- x) 1+ x + x2 +............+ xn + xn+1 ອຬຈຂຌໄ n ຽຑໃ ຌິ ແຈຑໄ ສິ ູຈ຾ຍຍຉຄໄ ຂຬຄກຳຌຍວກ, ຂຌໄ ຋ຳຬຈິ 1 ຾ລະ ຍຳຈກຳໄ ວ a 1+ a + a2 + ......... + an = 1- an+1 ຽຑຳະສຳລຍ a  K ຽອຳ຺ ຅ະແຈໄ 1- a ( )1 = (1- a) 1+ a + a2 +..........+ an + an+1 ກຳຌຑະຫຸຑຈ຺ ເຫ຋ໄ ະວຑີ ຈ຺ ຌງິ ຳມ: ກຳຌຫຳຌຑະຫຸຑຈ຺ Px ຈວໄ ງ x  a ສຳມຳຈຂຼຌເຌອູຍອໃ ຳຄລວມຈໃ ຄຌີໄ 193

ຑຈຶ ຆະ຃ະຌຈິ ຆຌໄ ສູຄ P(x) = (x - a)Q(x)+ R Q( x ) ຾ມໃ ຌຏຌ຺ ຋ໃແີ ຈອໄ ຍ຅ຳກກຳຌຫຳຌ R ຾ມໃ ຌຉວ຺ ຽສຈ ຽຫຌແຈວໄ ໃ ຳ (1) p(a) = (a - a)Q(a)+ R = R (2) px ຫຳຌຂຳຈ x  a ກຉໃ ຽມໃ ຬືຼ pa  0 ຉວ຺ ຢໃ ຳຄ 1: px  xn  an ຫຳຌຂະໜຳຈ x  a ໝຳງຽຊຄິ ວໃ ຳ ( )P (x) = (x - a) xn- 1 + axn- 2 + a2xn- 3 +............ + an- 2x + an- 1 ຉວ຺ ຢໃ ຳຄ 2: P(x) = x2 - 2x +1 ຫຳຌຂຳຈຈວໄ ງ x 1 ຽອຳ຺ ກວຈຽຍໃ ຄິ P(1) =1- 2 +1 = 0 ໝຳງຽຊຄິ ຉວ຺ ຽສຈຽ຋ໃ ຳ຺ ສູຌ, ແຈຏໄ ຌ຺ ຫຳຌ x2  2x 1  x 1x 1  0 ຉວ຺ ຢໃ ຳຄ 3: P(x) = 2x2 - 3x + 5 ຫຳຌຈວໄ ງ x +1 ຽອຳ຺ ກວຈຽຍໃ ຄິ ວໃ ຳ p1  2  3  5  10 ແຈຏໄ ຌ຺ ຫຳຌ 2x2 - 3x + 5 = (x +1)(2x - 5)+10 ຉວ຺ ຢໃ ຳຄ 4: Px  2x3  2x2  x 1 ຫຳຌຈວໄ ງ x  3 ຽອຳ຺ ກວຈຽຍໃ ຄິ ວໃ ຳ p3  54 18  3 1  38 ( )ແຈຏໄ ຌ຺ ຫຳຌ 2x3 - 2x2 + x - 1 = (x - 3) 2x2 + 4x +13 + 38 , p3  0 ໝຳງ຃ວຳມວໃ ຳ Px ຍໃ ຫຳຌຂຳຈຈວໄ ງ x  3  3 ກຳຌຫຳຌຑະຫຸຑຈ຺ ເຫ຋ໄ ະວຑີ ຈ຺ x  a ຈວໄ ງວ຋ິ ອີ ຬກຽຌ.ີ ກຳຌຫຳຌຑະຫຸຑຈ຺ A(x) = a0 + a1x + a2x +...+ anxn ຈວໄ ງ x  a ສຳມຳຈ ຎະຉຍິ ຈຉຳມວ຋ິ ີ ອຬກຽຌແີ ຈຈໄ ໃ ຄຌ:ີໄ 1. ຂຼຌສຳຎະສຈິ ຋ຸກໂຉວ຺ ຂຬຄ Ax ຽຆໃ ຌ : an, an-1, an- 2..., a0 ລຄ຺ ເຌ຾ຊວ຋ໜີ ໃ ຄຶ 2. ຂຼຌສຳຎະສຈິ a ຾ລະ an ລຄ຺ ເສໃ ຉຌ຺ໄ ຾ຊວ຋ສີ ຬຄ. 3. ຃ຈິ ແລໃ ຃ໃ ຳຂຬຄ bn- 1,bn- 2,...,b0 = R bn- 1 = aan + an- 1 bn- 2 = abn- 1 + an- 2 b0 = R = ab1 + a0 194

ຑຈຶ ຆະ຃ະຌຈິ ຆຌໄ ສູຄ 4. ຏຌ຺ ຋ໃ ແີ ຈອໄ ຍ຅ຳກກຳຌຫຳຌສຳມຳຈຂຼຌແຈຈໄ ໃ ຄຌີໄ ( )A(x) = (x - a) anxn- 1 + bn- 1xn- 2 +...+ b1 + R an an-1 an-2... a1 a0 a an b n-1 bn-2... b1 R ຉວ຺ ຢໃ ຳຄ 1: ຅ໃ ຄ຺ ຫຳຌ 4x3 - x5 + 32 - 8x2 ຈວໄ ງ x + 2 ວ຋ິ ຾ີ ກ:ໄ ຽອຳ຺ ຅ຈລຼຄຑະຫຸຑຈ຺ ກໃ ຬຌ : - x5 + 4x3 - 8x2 + 32 1 0 48 0 32 2 1 2 0 8 16 0 ( )ຈໃ ຄຌຌໄ 4x3 - x5 + 32 - 8x2 = (x + 2) - x4 + 2x3 - 8x +16 ຉວ຺ ຢໃ ຳຄ 2: ຅ໃ ຄ຺ ຫຳຌ x5 +8x4 + 20x2 - 15x2 + 8 ຈວໄ ງ x  5 1 23 42 5 1 3 18 86 432 ( )ຈໃ ຄຌຌໄ 4x4 - 2x3 + 3x2 - 4x + 2 = (x - 5) x3 + 3x2 +18x + 86 + 432 ( )ກວຈ຃ຌຼື (x - 5) x3 + 3x2 +18x + 86 + 432 = x4 - 2x3 + 3x2 - 4x + 2 ຉວ຺ ຢໃ ຳຄ 3: ຫຳຌ x5 +8x4 + 20x2 - 15x2 + 8 ຈວໄ ງ x + 3 1 8 0 20 15 8 3 1 5 15 65 210 638 ( )ຈໃ ຄຌຌໄ x5 + 8x4 + 20x2 - 15x + 8 = (x + 3) x4 - 5x3 - 15x2 + 65x - 210 + 638 ຉວ຺ ຢໃ ຳຄ 4: ຫຳຌ x3  8 ຈວໄ ງ x  2 10 08 2 12 4 0 ( )ຈໃ ຄຌຌໄ x3  8 = (x - 2) x2 + 2x + 4 195

ຑຈຶ ຆະ຃ະຌຈິ ຆຌໄ ສູຄ 4 ຽສຈສໃ ວຌຎ຺ກກະຉິ ຾ລະ ກຳຌ຾ງກຽສຈສໃ ວຌຎ຺ກກະຉຽິ ຎຌຽສຈສໃ ວຌ ຽສຈສໃ ວຌຎ຺ກກະຉ:ິ ຌງິ ຳມ: K ຾ມໃ ຌ຋ໃ ຄ຺ ຬຌໜໃ ຄຶ A ຾ລະ B ຾ມໃ ຌຑະຫຸຑຈ຺ ເຌ຋ໃ ຄ຺ K ຆໃ ຄຶ ວໃ ຳ B  0 ; A , AB 1 ຽຬຌີໄ ວໃ ຳ B ຽສຈສໃ ວຌຎ຺ກກະຉເິ ຌ຋ໃ ຄ຺ K. ຉວ຺ ຢໃ ຳຄ 1: ກ. x2 + 2x - 1 ຂ. x3 - 2x2 + 5x - 9 ຃. 2x4 - x3 - 6x2 + 4x - 8 x +1 x2 - 2 x2 - x + 2 (ຑະຫຸຑຈ຺ ຫຳຌຈວໄ ງຑະຫຸຑຈ຺ ຽຬຌີໄ ວໃ ຳ ຽສຈສໃ ວຌຎ຺ກກະຉ)ິ ຃ຸຌລກສະຌະ: A, B,C, D ລວໄ ຌ຾ຉໃ ຾ມໃ ຌຑະຫຸຑຈ຺ ຋ໃ ຢີ ໃ ູເຌ Kx ; A  C  AD  BC BD ( )( )x2 + x +1 x3 - 1 x +1 = x2 - 1 ຉວ຺ ຢໃ ຳຄ 2: ຽຑຳະວໃ ຳ x2 + x +1 x2 - 1 = x4 + x3 - x - 1  ຾ລະ x3 1 x 1  x4  x3  x 1 ຉວ຺ ຢໃ ຳຄ 3: x2 - x +1 x3 +1 x +1 = (x +1)2 ( )x3 +1 (x +1) = x4 + x3 + x +1 ( )x2 - x +1 (x +1)2 = x4 + x3 + x +1 ກຳຌ຾ງກຽສຈສໃ ວຌຎ຺ກກະຉຽິ ຎຌຽສຈສໃ ວຌ: ຏຌ຺ ຫຳຌຂຬຄສຬຄຑະຫຸຑຈ຺ ເຌອູຍອໃ ຳຄ A ຽຆໃ ຄິ ວໃ ຳ d  A  d B ຽຆໃ ຄິ ມຆີ ໃ ວືຼ ໃ ຳຽສຈສໃ ວຌ ຎ຺ກກະຉ,ິ B ຽສຈສໃ ວຌຄໃຳງຈຳງ຾ມໃ ຌຽສຈສໃ ວຌຎ຺ກກະຉ຋ິ ໃ ມີ ອີ ູຍອໃ ຳຄ: A 1. x- a 2. A ; mÎ (x - a)m 1 3. x2 + Px + q 4. A ; p2  4q  0 x2  px  q 196

ຑຈຶ ຆະ຃ະຌຈິ ຆຌໄ ສູຄ  Ax  B ; p2  4q  0 ; m Iˆ 5. x2  px  q m ກຳຌ຾ງກຽສຈສໃ ວຌຎ຺ກກະຉິ A ເຫກໄ ຳງຽຎຌຽສຈສໃ ວຌຄໃຳງຈຳງ຾ມໃ ຌຈຳຽຌຌີ ຉຳມ຾ຉໃ ລະອູຍ຾ຍຍຂຬຄ B B ຈໃ ຄຌ:ີໄ ກລະຌ຋ີ ີ 1: ຊຳໄ d  B  n ຾ລະ B ມີ n ອຳກ຃ຼື : a1, a2,...,an ຋ໃ ຽີ ຎຌ຅ຳຌວຌ຅ຄິ ຉໃ ຳຄກຌ ເຌກລະຌຌີ ີໄ A B ສຳມຳຈ຾ງກຬຬກຽຎຌຽສຈສໃ ວຌຄໃຳງຈຳງແຈອໄ ູຍອໃ ຳຄຈໃ ຄຌ:ີໄ A = A1 + A2 + ... + An B x - a1 x - a2 x - an ຉວ຺ ຢໃ ຳຄ 1: ຾ງກ x2 +1 ຾ງກຽຎຌສໃ ວຌຄໃຳງຈຳງ x (x - 1)(x +1) ວ຋ິ ຾ີ ກ:ໄ x2 +1 = A1 + A2 + A3 = (x - 1)(x +1) A1 + x (x +1) A2 + x (x - 1) A3 x x- 1 x +1 x (x - 1)(x +1) x(x - 1)(x +1) ຅ຳກ຃ຸຌລກສະຌະຂຬຄຽສຈສໃ ວຌ຅ະແຈໄ x +1 = (x - 1)(x +1) A1 + x(x +1) A2 + x (x - 1) A3 ຅ຳກສມ຺ ຏຌ຺ ສຸຈ຋ຳໄ ງຽຫຌແຈຈໄ ໃ ຄຌ:ີໄ ຽມໃ ຬືຼ x  0 . ຅ະແຈ ໄ 1  A1 ຾ລະ ຊຬຌແຈ ໄ A1  1 ຽມໃ ຬືຼ x  1 . ຅ະແຈ ໄ 2  2A2 ຾ລະ ຊຬຌແຈ ໄ A2  1 ຽມໃ ຬຼື x  1 . ຅ະແຈ ໄ 2  2A3 ຾ລະ ຊຬຌແຈ ໄ A3  1 ຈໃ ຄຌຌໄ x2 +1 -1 1 1 x (x - 1)(x +1) = ++ x x - 1 x +1 ກລະຌ຋ິ ີ 2: ຊຳໄ ອຳກ຋ຄໝຈ຺ ຂຬຄ B ຽຎຌ຅ຳຌວຌ຅ຄິ ຾ຉໃ ມຍີ ຳຄອຳກຆຬໄ ຌກຌຽຆໃ ຌ: B  x  ax  b2 x  c3 ເຌກລະຌຌີ ຽີໄ ສຈສໃ ວຌ A x  ax  b2 x  c3 ສຳມຳຈຂຼຌຽຎຌຽສຈສໃ ວຌຄໃຳງຈຳງເຌອູຍອໃ ຳຄຈໃ ຄຌ:ີໄ (x - A c)3 = A1 + A2 + (x A3 + A4 + ( x A5 + (x A6 )3 x- a x- b x- c - c)2 -c a)(x - b)2 (x - - b)2 197

ຑຈຶ ຆະ຃ະຌຈິ ຆຌໄ ສູຄ ຉວ຺ ຢໃ ຳຄ 2: ຾ງກຽສຈສໃ ວຌ 2x2 - 3x + 4 ຬຬກຽຎຌຽສຈສໃ ວຌຄໃຳງຈຳງ (x +1)(x - 2)2 ວ຋ິ ຾ີ ກ:ໄ 2x2 - 3x + 4 = A1 + A2 2 + (x A3 x +1 x- (x +1)(x - 2)2 - 2)2 = (x - 2)2 A1 + (x +1)(x - 2) A2 + (x +1) A3 (x +1)(x - 2)2 ຅ຳກຌ຅ີໄ ໃ ຄຶ ແຈໄ 2x2 - 3x + 4 = (x - 2)2 A1 + (x +1)(x - 2) A2 + (x +1) A3 ຽມໃ ຬືຼ x  1 ຅ະແຈ ໄ 9  9A1 ຾ລະ ຊຬຌແຈ ໄ A1  1 ຽມໃ ຬືຼ x  2 ຅ະແຈ ໄ 6  3A3 ຾ລະ ຊຬຌແຈ ໄ A3  2 ຽມໃ ຬຼື x  0 ຅ະແຈ ໄ 4 = - 2A2 + 6 ຾ລະ ຊຬຌແຈ ໄ A2  1 ຈໃ ຄຌຌໄ 2x2 - 3x + 4 1 1 2 = + + (x +1)(x - 2)2 x +1 x- 2 (x - 2)2 ກລະຌ຋ີ ີ 3: ຊຳໄ B ມອີ ຳກຽຎຌ຅ຳຌວຌສຌ຺ ຾ຉກຉໃ ຳຄກຌຽຆໃ ຌ ( )B = (x - a) x2 + bx + c ຽຆໃ ຄີ b2  4c  0 ເຌກລະຌຌີ ີໄ A ສຳມຳຈ຾ງກຬຬກຽຎຌ B ຽສຈສໃ ວຌຄໃຳງຈຳງແຈເໄ ຌອູຍອໃ ຳຄຈໃ ຄຌີໄ : A = A1 + A2 x + A3 B x- a x2 + bx + c ຉວ຺ ຢໃ ຳຄ 3: ຾ງກ 3x2 + x - 2 ຬຬກຽຎຌຽສຈສໃ ວຌຄໃຳງຈຳງ (x - 1)(x2 +1) ( )3x2 + x - 2 ( ) ( )(x - 1) x2 +1 ວ຋ິ ຾ີ ກ:ໄ = A1 + A2 x + A3 = x2 +1 A1 + (x - 1)(A2x + A3 ) x- 1 x2 +1 (x - 1) x2 +1 ( )຅ຳກຌແີໄ ຈໄ 3x2 + x - 2 = x2 +1 A1 + (x - 1)(A2x + A3 ) ຽມໃ ຬຼື x  1 ຅ະແຈ ໄ 2  2A1 ຾ລະ ຊຬຌແຈ ໄ A1  1 ຽມໃ ຬືຼ x  0 ຅ະແຈ ໄ  2  1 A3 ຾ລະ ຊຬຌແຈ ໄ A3  3 ຽມໃ ຬຼື x  1 ຅ະແຈໄ 0 = 2 - 2(- A2 + 3) ຾ລະ ຊຬຌແຈໄ A2  2 3x2 + x - 2 1 2x - 3 ( )ຈໃ ຄຌຌໄ = + (x - 1) x2 +1 x- 1 x2 +1 198

ຑຈຶ ຆະ຃ະຌຈິ ຆຌໄ ສູຄ ກລະຌ຋ີ ີ 4: ຊຳໄ B ມອີ ຳກຽຎຌ຅ຳຌວຌສຍຆຬໄ ຌກຌ ຽຆໃ ຌ ( )B = ax2 + bx + c 3 ຽຆໃ ຄິ b2  4ac  0 ເຌກລະຌຌີ ີໄ A ສຳມຳຈ຾ງກຬຬກຽຎຌຽສຈສໃ ວຌຄໃຳງຈຳງແຈເໄ ຌອູຍອໃ ຳຄຈໃ ຄຌ:ີໄ B ຉວ຺ ຢໃ ຳຄ 4: ຾ງກ x3 + 4x2 - 4x - 1 ຾ງກຬຬກຽຎຌຽສຈສໃ ວຌຄໃຳງຈຳງ ( )x2 +1 2 ( ) ( ) ( ) ( )ວ຋ິ ຾ີ ກ:ໄ ( )x3 + 4x2 - 4x - 1 = A1x + B1 + A2 xB2 = x2 +1 A1x + B1 + A2x + B2 x2 +1 2 x2 +1 x2 +1 2 x2 +1 2 ( )຅ຳກຌແີໄ ຈ ໄ x3 + 4x2 - 4x - 1 = x2 +1 (A1x + B1)+ A2x + B2 = A1x3 + B1x2 + (A1 + A2 ) x + (B1 + B2 ) ຅ຳກກຳຌສມ຺ ຋ຼຍສຳຎະສຈິ ຂຬຄຑະຫຸຑຈ຺ ຋ໃ ຢີ ໃ ູຽຍຬືຼໄ ຄຆຳໄ ງກຍສຳຎະສຈິ ຑະຫຸຑຈ຺ ຋ໃ ຢີ ໃ ູຽຍຬຼືໄ ຄຂວຳຂຬຄສະຽໝີ ຏຌ຺ ຂຳໄ ຄຽ຋ຄິ ຌ຅ີໄ ໃ ຄຶ ແຈ ໄ A1 =1 ; B1 = 4 ; 1+ A2 = - 4 ຾ລະ ຊຬຌແຈ ໄ A2  5 4 + B2 = - 1 ຾ລະ ຊຬຌແຈ ໄ B2  5 ຈໃ ຄຌຌໄ x3 + 4x2 - 4x - 1 x + 4 - 5x - 5 = x2 +1 + x2 +1 2 ( ) ( )x2 +1 2 199

ຑຈຶ ຆະ຃ະຌຈິ ຆຌໄ ສູຄ ຍຈ຺ ຽຐິກຫຈ 13 1. ຅ໃ ຄ຺ ຾ງກຽສຈສໃ ວຌລໃ ຸມຌຬີໄ ຬກຽຎຌຽສຈສໃ ວຌຄໃຳງຈຳງ 1) x 6) x x  2x 1 x  2x 1x 1 2) 1 3x2  x  2 x2 x 1 x 1 x2 1  7) 3x2  x  2 3x2  2 x 1 x2 1 xx 1 x2 1  3)  8) 4) x2 1 9) 1 xx 1x 1 x2 x 12 5) x 1 x  2x 1x  3 10) x2 x 1x  4 200