without actual division prove that x^4-5x^3+8x^2-10x+12 is divisible by x^2-5x+6
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Step-by-step explanation:
suppose x⁴-5x³+8x²-10x+12 = f(x)
x²-5x+6 = 0
=> x²-(2+3)x+6 = 0
=> x²-2x-3x+6 = 0
=> x(x-2)-3(x-2) = 0
=> (x-2)(x-3) = 0
2 and 3 are the roots of x²-5x+6
if x⁴-5x³+8x²-10x+12 is divisible by x²-5x+6 then f(2) and f(3) = 0
f(2) = (2)⁴-(5*(2)³)+(8*(2)²)-(10*2)+12
= 16-(5*8)+(8*4)-20+12
= 16-40+32-20+12
= 60-60
= 0
f(3) = (3)⁴-(5*(3)³)+(8*(3)²)-(10*3)+12
= 81-(5*27)+(8*9)-30+12
= 81-135+72-30+12
= 165-165
= 0
this is proved that x⁴-5x³+8x²-10x+12 is divisible by x²-5x+6
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