Question

Let R₁ and R₂ be the remainders when the polynomials x³-6x²+2x-k and kx³+12x²+14x-3 are divided by (1-2x) and (2x+1) respectively. If R₁ - R₂ = 25/8, find the value of k.​

Answer 1

`\large\underline{\sf{Given \: info-}}`

When, p(x) = x^3 - 6x^2 + 2x - k, is divided by (1 - 2x), Remainder = R1

When, g(x) = kx^3 + 12x^2 + 14x - 3, is divided by (2x + 1), Remainder = R2

R1 - R2 = 25/8

• Find the value of k.

`\large\underline{\sf{Solution-}}`

We are given that,

`\sf\longmapsto x^3-6x^2+2x-k/(1-2x)=R_1`

And,

`\sf\longmapsto kx^3+12x^2+14x-3/(2x+1)=R_2`

Taking p(x) first,

We have,

`\sf\longmapsto x^3-6x^2+2x-k/(1-2x)=R_1`

That is,

`\sf\longmapsto p(x)/(1-2x)=R_1`

Let, 1 - 2x = 0.

So, x = 1/2

As,

`\sf\longmapsto p(x)/(1-2x)=R_1`

So, by Factor Theorem,

`\sf\longmapsto p\left((1)/(2)\right)=R_1`

So,

`\sf\longmapsto\left((1)/(2)\right)^3-6\left((1)/(2)\right)^2+2\left((1)/(2)\right)-k=R_1`

`\sf\longmapsto(1)/(8)-6\!\!\!/^3\left((1)/(4\!\!\!/_2)\right)+2\!\!\!/\left((1)/(2\!\!\!/)\right)-k=R_1`

So,

`\sf\longmapsto (1)/(8)-(3)/(2)+1-k=R_1`

Taking LCM,

`\sf\longmapsto (1-3(4)+1(8)-k(8))/(8)=R_1`

`\sf\longmapsto (1-12+8+8k)/(8)=R_1`

So,

`\sf\longmapsto (-3-8k)/(8)=R_1 - - - -(1)`

Taking g(x) now,

We have,

`\sf\longmapsto kx^3+12x^2+14x-3/(2x+1)=R_2`

That is,

`\sf\longmapsto g(x)/(2x+1)=R_2`

Let, 2x + 1 = 0.

So, x = -1/2

As,

`\sf\longmapsto g(x)/(2x+1)=R_2`

So, by Factor Theorem,

`\sf\longmapsto g\left((-1)/(2)\right)= R_2`

So,

`\sf\longmapsto k\left((-1)/(2)\right)^3+12\left((-1)/(2)\right)^2+14\left((-1)/(2)\right)-3= R_2`

`\sf\longmapsto k(-1)/(8)+12\!\!\!/^3\left((1)/(4\!\!\!/)\right)+14\!\!\!\!\!/^7\left((-1)/(2\!\!\!/)\right)-3= R_2`

So,

`\sf\longmapsto (-k)/(8)+3\!\!\!/-7-3\!\!\!/= R_2`

`\sf\longmapsto (-k)/(8)-7= R_2`

Taking LCM,

`\sf\longmapsto (-k-7(8))/(8)= R_2`

So,

`\sf\longmapsto (-k-56)/(8)= R_2 - - - -(2)`

Now, we are also given that,

`\sf\longmapsto R_1-R_2=(25)/(8)`

From (1) & (2),

`\sf\longmapsto R_1-R_2=(25)/(8)`

`\sf\longmapsto (-3-8k)/(8) - (-k-56)/(8) =(25)/(8)`

Combining fractions,

`\sf\longmapsto (-3-8k+k+56)/(8\!\!\!/)=(25)/(8\!\!\!)`

`\sf\longmapsto 53-7k=25`

`\sf\longmapsto 53-25=7k`

`\sf\longmapsto 7k=28`

So,

`\sf\longmapsto k=(28\!\!\!\!\!/^(\:\:4))/(7\!\!\!/)`

Hence,

`\longmapsto\bf k=4`

Therefore, the value of k is 4.