Question

On dividing 3x2-x3-3x+5 by a polynomial g(x),the quotient and remainder were x-2 and 3 respectively.find the polynomial g(x).

Answer 1

Answer:

g(x) = -x² + x - 1

Step-by-step explanation:

Hi,

Given the quotient , q(x) = (x-2)

remainder, r(x) = 3

polynomial p(x) = -x³ + 3x² - 3x + 5

But we now, by using Euclid's algorithm

p(x) = g(x)*q(x) + r(x).

Since q(x) is of degree 1, g(x) should be a polynomial of degree 2 since

the given polynomial is of degree 3.

Let g(x) be of form ax² + bx + c, now we know that

p(x) = -x³ + 3x² - 3x + 5 = (ax² + bx + c)(x-2) + 3

= ax³ +(-2a + b) x² + (-2b + c)x + 3 -2c

Now comparing the coefficient of x³ on both sides, we get

a = -1

Comparing the coefficient of x², we get

-2a + b = 3

=> 2 + b = 3

=> b = 1

Comparing the coefficient of x, we get

-2b + c =  -3

=> -2 + c = -3

=> c = -1

Thu g(x) = -x² + x - 1.

Hope, it helped !