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in Mathematics by (106k points)

What must be added to 3x3 + x2 - 22x + 9 so that the result is exactly divisible by 3x2 + 7x - 6. 

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Best answer

Let p(x) = 3x3 + x2 - 22x + 9 and q(x) = 3x2 + 7x - 6

We know if p(x) is divided by q(x) which is quadratic polynomial therefore if p(x) is not exactly divisible by q(x) then the remainder be r(x) and degree of r(x) is less than q(x) (or Divisor) 

∴By long division method 

Let we added ax + b (linear polynomial) is p(x), so that p(x) + ax + b is exactly divisible by 3x2 + 7x - 6.

Hence p(X) + ax + b = s(x) = 3x3 + x2 - 22x + 9 ax + b 

= 3x3 + x2 - x(22 - a) + (9 + b) 

Hence, x(a – 2 + b – 3 = 0 . x + 0) 

⇒ a - 2 = 0 & b - 3 = 0 

⇒ a = 2 or b = 3 

Hence, if we add ax + b or 2x + 3 in p(x) then it is exactly divisible by 3x2 + 7x - 6. 

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