show that x-2 is a factor of f(x) = 2x cube -3x square - 17x + 30 and hence factorise

f (x )

fx = 2x3 - 3x2 - 17x + 30If x - 2 is a factor of fx then f2 must be zeroNow, f2 = 2 23 - 3 22 - 17 × 2 + 30     f2 = 2× 8 - 3 × 4 - 34 + 30     f2 = 16 - 12 - 34 + 30     f2 = 0Since f2 = 0Hence, x - 2 is a factor of fxNow,  fx = 2x3 - 3x2 - 17x + 30                  = 2x3 - 4x2 + x2 - 2x - 15x + 30                  = 2x2 x - 2 + x x - 2 - 15 x - 2                  =x - 2 2x2 + x - 15                  = x - 2 2x2 + 6x - 5x - 15                  = x - 2 2x x + 3 -5 x + 3                  = x - 2 x + 3 2x - 5                  = x - 2 x + 3 2x - 5

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Question: To show that x-2 is a factor of f(x)= 2x3-3x2-17x+30 and also to factorise it

Answer : we have to showthat x-2 is a factor of f(x)= 2x3-3x2-17x+30

Let's put x =2 in f(x) , if the remainder comes out to be 0 then x-2 will be the factor of f(x)

f(2)=2(2)3-3(2)2-17(2)+30

f(2)=2(8)-3(4)-34+30 = 16-12-34+30 = -46+46 = 0

hence x-2 is a factor of f(x)

Now to factorise f(x)

now we do long division of2x3-3x2-17x+30 , which is divided by x-2

we get 2x2+x-15

f(x)= (x-2)(2x2+x-15)

f(x)= (x-2)(2x2+6x-5x-15)

f(x)= (x-2)[2x(x+3)-5(x+3)]

f(x)= (x-2)(2x-5)(x+3) - Answer

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