Question

On dividing x^3 - 3x² + x + 2 by a polynomial g(x)
the quotient and the remainder are x-2 and -2x+4
respectively find g(x)

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Answer 1

Step-by-step explanation:

Given :-

• On dividing x³ - 3x² + x + 2 by a polynomial g(x) the quotient and the remainder are x-2 and -2x+4 respectively.

To Find :-

Find g(x) ?

Solution :-

Given polynomial is x³ - 3x² + x + 2

Let p(x) = x³ - 3x² + x + 2

The divisor = g(x)

The Quotient = q(x) = x-2

Remainder = r(x) = -2x+4

We know that

Fundamental Theorem on Polynomials is

p(x) = g(x)×q(x) + r(x)

=> x³ - 3x² + x + 2 = g(x) × (x-2) + (-2x+4)

=> x³ - 3x² + x + 2 +2x-4 = g(x) × (x-2)

=> x³ -3x²+3x-2 = g(x) × (x-2)

=> g(x) = x³ -3x²+3x-2 ÷ (x-2)

=> g(x) = (x³ -3x²+3x-2 )/(x-2)

x-2 ) x³-3x²+3x-2 ( x² -x +1

x³-2x²

(-) (+)

__________

-x² +3x

-x² +2x

(+) (-)

___________

x -2

x -2

(-) (+)

____________

0

____________

=> g(x) = (x³ -3x²+3x-2 )/(x-2)

=> g(x) = x²-x+1

Answer:-

g(x) for the given problem is x²-x+1

Used formulae:-

→Fundamental Theorem on Polynomials is p(x) = g(x)×q(x) + r(x)