Math, asked by ravidde5, 7 days ago

TELL WITH CORRECT ANSWER FASTIY WITH CORRECT EXPLANATION

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Answered by devanshu1234321

119

QUESTION:-

$\sf\;If\;x-2\;is\;common\;factor\;of\;expression\;x^2+ax+b\;and \;x^2+cx+d\;then\;find\frac{b-d}{c-a}$

EXPLANATION:-

If x-2 is a factor then x=2 is root of the given expressions

So let:-

p(x)=x²+ax+b

and f(x)=x²+cx+d

Now let's find value of p(2) and f(2)

p(2)=4+2a+b

and,

f(2)=4+2c+d

Since 2 is factor of p(2) and f(2) we can write :-

4+2a+b=0--eq-1 and 4+2c+d=0--eq-1

EQUATING THEM:-

WE get:-

4+2a+b=4+2c+d

→2a+b=2c+d

→2a-2c=b-d

→2(a-c)=b-d

→2=b-d/a-c

So the value fo b-d/a-c is 2

Question:-

If (x - 2) is a common factor of the expression x² + ax + b and x² + cx + d then (b - d)/(c - a) =

Answer:-

Given:-

(x - 2) is a common factor of x² + ax + b and x² + cx + d.

Let p(x) = x² + ax + b and q(x) = x² + cx + d

That means, when p(x) and q(x) are divided by x - 2 leave 0 as the remainder.

Using Factor Theorem,

⟹ x - 2 = 0

⟹ x = 2

x = 2 is the common factor. So, p(2) = q(2)

⟹ p(2) = q(2)

⟹ (2)² + a * 2 + b = (2)² + c(2) + d

⟹ 4 + 2a + b = 4 + 2c + d

⟹ 2a + b = 2c + d

⟹ b - d = 2c - 2a

⟹ (b - d) = 2(c - a)

⟹ (b - d) / (c - a) = 2

∴ Required answer is 2 (Option - D).

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