How do i find k when x^3+kx^2+2x-3 is divided by X+2 and the remainder is 1

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hala718's profile pic

hala718 | High School Teacher | (Level 1) Educator Emeritus

Posted on

let f(x) = x^3 + kx^2 + 2x -3

When we devide f(x) with (x+2) th result will be a function (let it be R(x) and a remainder of 1:

Then we could write:

==> f(x)= (x+2) * R(x) + 1

Let us substitute with x= -2:

==> f(-2) = (-2+2)* R(-2) + 1

==> f(-2) = 1.........(1)

But, f(x) = x^3 + kx^2 + 2x -3

Let us sustitute x=-2 in f(x) .

==> f(-2) = -8 + 4k - 4 -3 = 1

==> 4k -15 = 1

==> 4k = 16

==> k= 4

==> f(x) = x^3 + 4x^2 + 2x - 3

==> f(x) = (x+2)(x^2 + 2x -2) + 1

neela's profile pic

neela | High School Teacher | (Level 3) Valedictorian

Posted on

x^3+kx^2+2x-3 gives a remainder 1 when divided by x+2 gives a remainder 1.

We do the actual operation of division:

x+2) x^3+kx^2 +2x-3( x^2

        x^3 +2x^2

------------------------

x+2)  (k-2)x^2 +2x ( (k-2)x

          (k-2)x^2 +2(k-2)x

--------------------------------------

x+2) (-2k+6)x          - 3   (     (-2k+6)

        (-2k+6)x - 2(-2k+6)

-----------------------------------

                 -4k  +  9 . But this should be equal to 1 as remainder

is  1 by data given.

So -4k+9 = 1.

-4k = 1-9 = -8. 

 k = -8/-4 = 2.

Thus if k = 2, then the given expression  would be x^3+2x^2+x-3  will give a remainder 1 when divided by x+2.

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