Question

If x²-3x+2 is a factor of the polynomial x⁴-ax³+b, then find the values of a and b

Answer 1

(x² -3x + 2) is a factor of x⁴ -ax³ + b ,

here ,
x² - 3x + 2
= x² -2x - x + 2
=(x -1)(x -2)

hence,
(x -1) and (x -2) are the factors of x⁴ - ax³ + b .

so, x = 1 and 2 are the zeros of x⁴ - ax³ + b

put x= 1 in x⁴ -ax³ + b

(1)⁴ -a(1)³ + b = 0

1 - a + b = 0

a - b = 1 -----(1)

now,
put x = 2 in x⁴ -ax³ + b

(2)⁴ - a(2)³ + b = 0

16 - 8a + b = 0

8a - b = 16 --------------(2)

solve equation (1) and (2)

8( b + 1) - b = 16
8b + 8 - b = 16

b = 8/7 and a = 15/7