Question

A polynomial f(x) with rational coefficients leaves remainder 15, when divided by(x – 3) and remainder 2x + 1, when divided by (x – 1)2. If ‘p’ is coefficient of x^2 of its remainder which will come out if f(x) is divided by (x – 3)(x – 1)^2 then find p.​

Answer 1

$\large\text{\underline{Let's begin:-}}$

Let us consider the identity,

$\hookrightarrow f(x)=(x-3)(x-1)^{2}Q(x)+R(x)$

which can be obtained by division. $\large\text{[1]}$

If an n-th degree polynomial divides another polynomial, the maximum degree of the remainder is n-1. $\large\text{[2]}$

$\large\text{\underline{Solution:-}}$

By remainder theorem, the first condition is $f(3)=15$.

(By $\large\text{[1]}$ and $\large\text{[2]}$)

The remainder of $f(x)$ by $(x-1)^{2}$ is equivalent to the remainder of $R(x)$ by $(x-1)^{2}$.

Since $2x+1$ is the remainder if $f(x)$ is divided by $(x-1)^{2}$, and as the maximum degree of $R(x)$ is 2,

$\hookrightarrow R(x)=p(x-1)^{2}+2x+1$

Now, according to $f(3)=15$, we get,

$\hookrightarrow f(3)=R(3)$

$\hookrightarrow R(3)=15$

$\hookrightarrow p\times(3-1)^{2}+2\times3+1=15$

$\hookrightarrow 4p+7=15\implies\therefore p=2$

$\large\text{\underline{Result:-}}$

The required value of $p$ is 2.

Answer 2

Answer:

Since function f(x) leaves remainder 15 when divided by x−3, therefore f(x) can be written as

f(x)=(x−3)l(x)+15 ...(1)

Also, f(x) leaves remainder 2x+1 when divided by (x−1)

2

.

Thus, f(x) can also be written as

f(x)=(x−1)

2

m(x)+2x+1 ...(2)

If R(x) be the remainder when f(x) is divided by (x−3)(x−1)

2

, then we may write

f(x)=(x−3)(x−1)

2

n(x)+R(x) ...(3)

Since (x−3)(x−1)

2

is a polynomial of degree three, the remainder has to be a polynomial of degree less than or equal to two.

Thus let R(x)=ax

2

+bx+c

From (1) and (3), we have

f(3)=15=R(3)⇒9a+3b+c=15 ...(4)

From (2) and (3), we have

f(1)=3=R(1)⇒a+b+c=3 ...(5)

From (2) and (3), we have

f

′

(1)=2=R

′

(1)⇒2a+b=2 ...(6)

Solving equation (4),(5) and (6), we get

a=2,b=−2,c=3

Step-by-step explanation:

Answer is 2x

2

−2x+3