Question

If ax² + bx + c and bx² + ax + c have a common factor x+1 then show that c = 0 and a = b ​

Answer 1

ax² + bx + c= (x+1)(ax+(b-a))

only if b-a=c

and

bx² + ax + c =(x+1)(bx+(a-b))

again if a-b=c

》》so a=b

and c=0

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Answer 2

Step-by-step explanation:

$\huge\bold\red{let}$

f(x) = ax² + bx + c and

g(x) = bx² + ax + c

( x + 1 ) is the common factor of f(x) and g(x)

★ If ( x + 1 ) is a factor of f(x) then the remainder f(-1) = 0

→ f(-1) = a(-1)² + b(-1) + c = 0

→ f(-1) = a × 1 - b + c = 0

→ f(-1) = a - b + c = 0

$\huge\boxed{\fcolorbox{cyan}{lightyellow}{f(-1)~=~a~+~c~=~b}}$ → (1)

★ If ( x + 1 ) is a factor of g(x) then the remainder g(-1) = 0

→ g(-1) = b(-1)² + a(-1) + c = 0

→ g(-1) = b × 1 - a + c = 0

→ g(-1) = b + c = a

$\huge\boxed{\fcolorbox{cyan}{orange}{g(-1)~=~a~=~b~+~c}}$ → (2)

substitute " a " value in equation (1)

a + c = b

( b + c ) + c = b

c + c = b - b

2c = 0

c = 0 / 2

$\huge\boxed{\fcolorbox{cyan}{purple}{c~=~0}}$

substitute " c " value in equation (2)

a = b + c

a = b + 0

$\huge\boxed{\fcolorbox{cyan}{yellow}{a~=~b}}$

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