Let $f (x) = 2 x^{2} + 5 x + 1$ . If we write $f (x)$ as
$f (x) = a (x + 1) (x - 2) + b (x - 2) (x - 1) + c (x - 1) (x + 1)$ for real numbers $a, b, c$ then

there are infinite number of choices for

a, b, c

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only one choice for

a

but infinite number of choices for

b

and

c

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exactly one choice for each of

a, b, c

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more than one but finite number of choices for

a, b, c

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Solution

The correct option is A exactly one choice for each of $a, b, c$
Comparing coefficients :

Comparing $x^{2}$ coefficient $\Rightarrow a + b + c = 2$

Comparing $x$ coefficient $\Rightarrow - a - 3 b = 5$

Comparing constants $\Rightarrow - 2 a + 2 b - c = 1$

There are 3 different equations with 3 unknowns and none of the equations are identical $\Rightarrow$ we get exactly one value for each $a, b, c$

$∴$ option 'C' is correct

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Let f(x)=2x2+5x+1. If we write f(x) asf(x)=a(x+1)(x−2)+b(x−2)(x−1)+c(x−1)(x+1) for real numbers a,b,c then

Let $f (x) = 2 x^{2} + 5 x + 1$ . If we write $f (x)$ as
$f (x) = a (x + 1) (x - 2) + b (x - 2) (x - 1) + c (x - 1) (x + 1)$ for real numbers $a, b, c$ then