Let P ( x )= x 2- cx - c. If P ( x ) is divided by ( x -2), the remainder is the same as when ( P ( x ))2 is divided by (x-2), then the sum of all possible values of c, is

Question

Let P ( x )= x 2- cx - c. If P ( x ) is divided by ( x -2), the remainder is the same as when ( P ( x ))2 is divided by (x-2), then the sum of all possible values of c, is (A) 1 (B) (5/3) (C) (1/3) (D) (7/3)

Tardigrade Admin · Accepted Answer

P(x)=(x-2) Q1+R1 (P(x))2=(x-2) Q2+R2 text Given, R1=R2 Given, R1=R2. ⇒ P (2)=( P (2))2 ⇒ P (2)=0 or 1 Hence, 4-3 c =0 or 1 ∴ c =(4/3) or 1 ⇒ Sum =(7/3).

Q. Let $P (x) = x^{2} - c x - c$ . If $P (x)$ is divided by $(x - 2)$ , the remainder is the same as when $(P (x))^{2}$ is divided by $(x - 2)$ , then the sum of all possible values of $c$ , is

1

$\frac{5}{3}$

$\frac{1}{3}$

$\frac{7}{3}$

$P (x) = (x - 2) Q_{1} + R_{1}$
$(P (x))^{2} = (x - 2) Q_{2} + R_{2}$
$Given, R_{1} = R_{2}$
Given, $R_{1} = R_{2}$ .
$\Rightarrow P (2) = (P (2))^{2} \Rightarrow P (2) = 0$ or 1
Hence, $4 - 3 c = 0$ or 1
$∴ c = \frac{4}{3}$ or $1 \Rightarrow$ Sum $= \frac{7}{3}$ .

Q. Let P(x)=x2−cx−c. If P(x) is divided by (x−2), the remainder is the same as when (P(x))2 is divided by (x−2), then the sum of all possible values of c, is

1

35​

31​

37​

P(x)=(x−2)Q1​+R1​ (P(x))2=(x−2)Q2​+R2​ Given, R1​=R2​ Given, R1​=R2​. ⇒P(2)=(P(2))2⇒P(2)=0 or 1 Hence, 4−3c=0 or 1 ∴c=34​ or 1⇒ Sum =37​.

Q. Let $P (x) = x^{2} - c x - c$ . If $P (x)$ is divided by $(x - 2)$ , the remainder is the same as when $(P (x))^{2}$ is divided by $(x - 2)$ , then the sum of all possible values of $c$ , is

$\frac{5}{3}$

$\frac{1}{3}$

$\frac{7}{3}$

$P (x) = (x - 2) Q_{1} + R_{1}$
$(P (x))^{2} = (x - 2) Q_{2} + R_{2}$
$Given, R_{1} = R_{2}$
Given, $R_{1} = R_{2}$ .
$\Rightarrow P (2) = (P (2))^{2} \Rightarrow P (2) = 0$ or 1
Hence, $4 - 3 c = 0$ or 1
$∴ c = \frac{4}{3}$ or $1 \Rightarrow$ Sum $= \frac{7}{3}$ .