2x^3+ax^2+b-2 has a factor (x+2) and leaves a remainder 7 when dividend by 2x-3
Answers
Answer:
Given polynomial
2x³ + ax² + bx - 2
(x + 2) is a factor of given polynomial
➞ x + 2 = 0
➞ x = - 2
Remainder = 0
2x³ + ax² + bx - 2 = 0
Put the value of x
➞ 2(-2)³ + a(-2)² + b(-2) - 2 = 0
➞ 2 × (-8) + 4a - 2b - 2 = 0
➞ -16 + 4a - 2b - 2 = 0
➞ 4a - 2b = 18
➞ 2(2a - b) = 18
➞ 2a - b = 9 ----(i)
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(2x - 3) is a factor of given polynomial
➞ 2x - 3 = 0
➞ x = 3/2
Put the value of x
Remainder = 7
2x³ + ax² + bx - 2 = 7
➞ 2(3/2)³ + a(3/2)² + 3b/2 - 2 = 7
➞ 2 × 81/8 + 9a/4 + 3b/2 = 7 + 2
➞ 27/4 + 9a/4 + 3b/2 = 9
➞ 27 + 9a + 6b/4 = 9
➞ 27 + 9a + 6b = 9 × 4
➞ 9a + 6b = 36 - 27
➞ 3(3a + 2b) = 9
➞ 3a + 2b = 3 ----(ii)
★ Multiply (i) by 2 and (ii) by 1
4a - 2b = 18
3a + 2b = 3
★ Add both the equations
➞ 4a - 2b + 3a + 2b = 18 + 3
➞ 7a = 21
➞ a = 21/7 = 3
★ Putting the value of a in eqⁿ (ii)
➞ 3a + 2b = 3
➞ 3 × 3 + 2b = 3
➞ 9 + 2b = 3
➞ 2b = 3 - 9
➞ 2b = - 6
➞ b = -6/2 = - 3
Hence,
Required value of a = 3
Required value of b = - 3