Question

If (x-1) is a factor of x⁴+2ax³-2x+b and a+b=4, then ab is equal to

Answer 1

Here , the following information is given -


(x-1) is a factor of x⁴+2ax³-2x+b and a+b = 4


To find -


Find the value of ab .


Solution -


Let us assume that -

f ( x ) = x⁴+2ax³-2x+b

g ( x ) = x - 1

Here , g ( x ) is a factor of f ( x ) .


According to the remainder Theorem -


When , f ( x ) is substituted by the value of The zero of g ( x ), the remainder is 0 .


Zero of g ( x ) -


=> ( x - 1 ) = 0


=> x = 1


=> f ( 1 ) = 0


=> 1⁴ + 2a1³ - 2× 1 + b


=> 1 + 2a - 2 + b


=> 2a + b - 1 = 0


=> 2a + b = 1 ......... Equation 1 .


We also have this given -


a+b=4 ........ Equation 2


Substituting this on Equation 1 ,


4 + a = 1


=> a = -3


=> b = 7


Required value -



=> ab


=> -3 × 7


=> -21 .


This is the required answer .


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Answer 2

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✒ If (x-1) is a factor of x⁴+2ax³-2x+b and a+b=4, then ab is equal to ?

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Given :-

• (x-1) is a factor of x⁴+2ax³-2x+b

• a+b= 4

To find :-

The value of ab

Solution :-

As, The quadratic equation is x⁴+2ax³-2x+b = 0

And x - 1 is its factor  ,

So,   x = 1 satisfy the equation

Put the value of x in eq ,

▶ x⁴+2ax³-2x+b = 0

i.e 1⁴+2×a×1³-2×1+b = 0

Or,  1 + 2a -2 +b = 0

Or,  2a + b -1 = 0

i.e  2a + b = 1       .......(1)

Since    a + b = 4      ......(2)

Adding 1 & 2 equation

2a + b = 1       .......(1)

a + b = 4      ......(2)

- - -

_______________

a = -3

_______________

➡ a = -3

Putting in (2)

➡ a + b = 4    

➡ -3 + b = 4

➡ b = 7

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The value of ab

ab = a × b

Where a = -3 , b =7

ab = -3×7

ab = -21 ✔

✏ Hence, The value of ab is -21 .

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