Question

Find the value of a and b so that the polynomial x³-4x²+ax+b is exactly divisible by x-2 as well as x+1.

Answer 1

By factor theorem:

Dividend= Divisor × Quotient + Remainder

So, to be completely divisible, Remainder= 0

x³ - 4x² + ax + b ÷ x - 2

x-2 = 0

x = 2

Substituting the value of x:

(2)³ - 4(2)² + a(2) + b = 0

8 - 16 + 2a + b = 0

2a + b - 8 = 0.................. Equation 1

x + 1 = 0

x = -1

Substituting the value of x:

(-1)³ - 4(-1)² + a(-1) + b = 0

-1 - 4 - a + b = 0

- a + b - 5 = 0.................... Equation 2

Subtracting equation 2 from 1

2a + b - 8 = 0

(-) -a + b - 5 = 0

3a. 0. -3. = 0

3a = 3

a = 1

Substituting the value of a in Equation 1

2 (1) + b - 8 = 0

b - 6 = 0

b = 6

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