Chia Da Thuc

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Ngày 26 tháng 9 năm 2009 1. Chứng minh rằng : a) a là nghiệm của đa thức f (x) khi và chỉ khi f (x) chia hết cho x − a. b) Dư của phép chia đa thức f (x) cho x − a là f (a). c) Dư của phép chia đa thức f (x) cho ax − b là f (b/a). Áp dụng : 1.1. Use the factor theorem to show : a) (x − 1) is a factor of f (x) = 4x3 − 3x2 − 1. b) (x + 3) is a factor of g(x) = 5x4 − 45x2 − 6x − 18. 1.2. Show that (x − 1) is a factor of x3 + 6x2 + 5x − 12 and hence factorise the expression completely. 1.3. Each of these expressions has a factor (x ± p). Find a value of p and hence factorise the expresstion completely : a) x3 − 10x2 + 19x + 30. b) x3 − 4x2 − 11x + 30. 1.4. Given that (x − 1) is a factor of 5x3 − 9x2 + 2x + a, find the value of a. 1.5. Given that (x + 3) is a factor of 6x3 − bx2 + 18, find the value of b. 1.6. Given that (x − 1) and (x + 1) are factors of px3 + qx2 − 3x − 7, find the value of p and q. 1.7. Find the remainder when : a) 4x3 − 5x2 + 7x + 1 is divided by (x − 2). b) −2x3 + 6x2 + 5x − 3 is divided by (x + 1). c) 4x4 − 4x2 + 8x − 1 is divided by (2x − 1). d) 64x3 + 32x2 − 16x + 9 is divided by (4x + 1). 1.8. When 2x3 − 3x2 − 2x + a is divided by (x − 1) the remainder is −4. Find the value of a.

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1.9. Show that (2x − 1) is a factor of 2x3 + 17x2 + 31x − 20 and hence factorise the expression completely. 1.10. g(x) = x3 + ax2 + 3x + 6. Given g(−1) = 2, find the remainder when g(x) is divided by (3x − 2). 1.11. The expression 2x3 − x2 + ax + b gives a remainder 14 when divided by (x−2) and a remainder -86 when divided by (x+3). Find the value of a and b. 1.12. The expression 3x3 + 2x2 − px + q is divisible by (x − 1) but leaves a remainder of 10 when divided by (x + 1). Find the value of a and b. 1.13. The remainder obtained when x3 + dx2 − 5x + 6 is divided by (x − 1) is twice the remainder obtained when the same expression is divided by (x + 1). Find the value of d. 2. Xác định a sao cho đa thức A chia hết cho đa thức B : a) A = 8x2 − 26x + a; B = 2x − 3. b) A = x3 − 13x + a; B = x2 + 4x + 3. 3. Cho A = 2a2 − b2 + ab − 5a + b + 2. a) Chứng minh a = b − 2 là nghiệm của đa thức A với biến là a. b) Phân tích A thành nhân tử. 4. Cho a = x2 − yz, b = y 2 − zx, c = z 2 − xy. Chứng minh ax + by + cz chia hết cho a + b + c. 5. Tìm các số nguyên x để các biểu thức sau nhận giá trị nguyên : 4x3 − 6x2 + 8x 2x − 1 3 4x − 3x2 + 2x − 83 B= x−3 A=

6. Thực hiện phép chia : a) [3(x − y)3 + 2(x − y)2 ] : (y − x)2 ; b) [10(x − y)4 − 12(y − x)5 − 4(x − y)6 ] : 2(y − x)4 .

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7. Cho P (x) = 3x3 − 2x2 + 5. Chia P (x) cho Q(x) 6= 0 được thương S(x) = 3x − 2 và dư R(x) = 3x + 3. Tìm Q(x). 8. Chứng minh rằng : a) Nếu ax + b = 0 ∀x thì a = b = 0. b) Nếu ax2 + bx + c = 0 ∀x thì a = b = c = 0. c) Em hãy tổng quát bài toán. 9. Tìm p và q để đa thức x5 − 7x4 + 15x2 + px + q chia hết cho đa thức x3 + 2x + 1. 10. Tìm a và b để đa thức x4 + x3 + ax2 + 4x + b chia hết cho đa thức x2 − 2x + 2.

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