A Polynomial has a remainder of 11 when it is divided by (x - 1) and a remainder of 4 when it is divided by..?

Question

A Polynomial has a remainder of 11 when it is divided by (x - 1) and a remainder of 4 when it is divided by (x+2). What is the remainder when this polynomial is divided by x*2 + x -2.

Answer 1

Let P be the polynomial. The remainder of the division of P by x*2 + x -2 = (x-1) (x +2) is a polynomial of degree 1, that is, a polynomial R of the form R(x) = a*x + b. So, there exists a polynomial Q such that P(x) = (x-1) (x +2) Q(x) + R(x). If we plug x =1 and x =-2 in this equation, we get P(1) = R(1) and P(-2) = R(-2). In addtion, we know P(1) is the remainder of the division of P by (x-1) and P(-2) is the remainder of the division of P by (x+2)., so that P(1) = 11 and P(-2) = 4. So, we have the system of linear equations:

a + b = 11
-2*a + b = 4 Multiplying the first equation by 2 and adding it to the second, we get 3b = 26 => b = 26/3. And a = 11 - b = 11 -26/3 = 7/3.

Therefore, your remainder is R(x) = 7/3 x + 26/3

Answer 2

P(x) = (x-1)*(x+2) *Q(x) +ax+b

You Know thatP(1) = 11 and P(-2)=4 .Sustituting in the above

11=a+b and 4= -2a+b Substracting 7=3a so a= 7/3 and b=26/3

The remainder is 7/3 *x +26/3

Answer 3

x3 + 3 x2 - 5x + 4 by utilizing x + one million? long branch: divide the foremost term (x^3) of the numerator by utilizing the 1st term (x) of the denominator x^3/x=x^2 then multiply that by utilizing the full denominator x^2(x+one million)=x^3+2x subtract x^2-5x+4 repeat x^2/x=x x(x+one million)=x^2+x subtract -6x+4 -6x/x= -6 -6(x+one million)= -6x-6 subtract 10 the rest is 10/(x+one million)