If x2-x-1-x2-2x-1 -A-B-x-1-C-x-12- then A - B is equal to

Explanation for the correct answer:(x^2+x+1)/(x^2+2x+1)=A+B/(x+1)+C/(x+1)^2By taking the LCM we get⇒(x^2+x+1)/(x+1)^2=A(x+1)^2+B(x+1)+C/(x+1)^2Multiplying the e ...

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Byju's Answer

Standard IX

Mathematics

Algebraic Identities

If x2+x+1/x2+...

Question

If $\frac{(x^{2} + x + 1)}{(x^{2} + 2 x + 1)}$ $= A + \frac{B}{(x + 1)} + \frac{C}{{(x + 1)}^{2}}$ , then $A - B$ is equal to

$4 C$

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$4 C + 1$

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$3 C$

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$2 C$

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Solution

The correct option is D
$2 C$

Explanation for the correct answer:
$\frac{(x^{2} + x + 1)}{(x^{2} + 2 x + 1)}$ $= A + \frac{B}{(x + 1)} + \frac{C}{{(x + 1)}^{2}}$
By taking the LCM we get
$\Rightarrow$ $\frac{(x^{2} + x + 1)}{{(x + 1)}^{2}}$ $= \frac{A {(x + 1)}^{2} + B (x + 1) + C}{{(x + 1)}^{2}}$
Multiplying the equation throughout by ${(x + 1)}^{2}$ we get
$\Rightarrow$ $(x^{2} + x + 1)$ $= A {(x + 1)}^{2} + B (x + 1) + C$
$\Rightarrow$ $(x^{2} + x + 1)$ $= A (x^{2} + 2 x + 1) + B (x + 1) + C$
$\Rightarrow$ $(x^{2} + x + 1)$ $= A x^{2} + (2 A + B) x + A + B + C$
On comparing the coefficients we get
$A = 1$ $. . . (i)$
$2 A + B = 1$ $. . . (i i)$
$A + B + C = 1$ $. . . (i i i)$
From (i) and (ii) we get
$2 (1) + B = 1$
$\Rightarrow$ $B = 1 - 2$
$\Rightarrow$ $B = - 1$
Substituting the value of $A, B$ in $(i i i)$ we get
$1 + (- 1) + C = 1$
$\Rightarrow$ $C = 1$
$\Rightarrow$ $A - B = 1 - (- 1)$
$\Rightarrow$ $A - B = 2 \times 1 = 2 C$
Hence, the value of $A - B$ for the given equation is $2 C$ , so option (D) is the correct answer.

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If (x2+x+1)(x2+2x+1) =A+Bx+1+Cx+12, then A-B is equal to

If $\frac{(x^{2} + x + 1)}{(x^{2} + 2 x + 1)}$ $= A + \frac{B}{(x + 1)} + \frac{C}{{(x + 1)}^{2}}$ , then $A - B$ is equal to