Question

find the value of a and b so that the polynomial x^4+ax^3-7x^2+8x+b is exactly divisible by (x+2) as well as (x+3)

Answer 1

Hey!!!

Good Afternoon

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Let p(x) = x⁴ + ax³ - 7x² + 8x + b

Required factors = (x + 2) and (x + 3)

Required Zeros = (-2) and (-3)

Thus let p(-2) = 0

=> (-2)⁴ + a(-2)³ - 7(-2)² + 8(-2) + b = 0

=> 16 - 8a - 28 - 16 + b = 0

=> 8a - b = -28

=> b = 8a + 28 --------(1)

Now p(-3) = 0

=> (-3)⁴ + a(-3)³ - 7(-3)² + 8(-3) + b = 0

=> 81 - 27a - 63 - 24 + b = 0

=> -27a + b = 6

=> b = 27a + 6 -----(2)

Equation (1) and (2)

=> 8a + 28 = 27a + 6

=> 19a = 22

=> a = 22/19

Using a in (1)

=> 8a + 28 = b

=> b = 8(22/19) + 28

=> b = 708/19

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Hope this helps ✌️