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The expression 4x3 – bx2 + x – c leaves remainders 0 and 30 when divided by x + 1 and 2x – 3 respectively. Calculate the values of b and c. Hence, factorise the expression completely. - ICSE Class 10 - Mathematics

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Question

The expression 4x3 – bx2 + x – c leaves remainders 0 and 30 when divided by x + 1 and 2x – 3 respectively. Calculate the values of b and c. Hence, factorise the expression completely. 

Solution

Let f (x)=4x^3-bx^2+x-c 

it is given that when f(x) is dividend by (x+1), the remainder is 0 

∴ `f(-1)=0` 

`4(-1)^3-b(1)^2+(-1)-c=0` 

`-4-b-1-c=0` 

`b+c+5=0 `........(1) 

It is given that when f(x) is divided by (2x-3) the remainder is 30. 

∴` f(3/2)=30 ` 

`4(3/2)^3 -b(3/2)^2+(3/2)-c =30 ` 

`27/2-(9b)/4+3/2-c=30 `

`54-9b+6-4c-120=0 `

`9b+4c+60=0 `  .............(2)  

Multiplying (1) by 4 and subtracting it from (2), we get, 

`5b+40=0 `

`b=-8 `

Substituting the value of b in (1), we get, 

`c=-5+8=3 ` 

Therefore, f(x) =4x^3+8x^x-3 

Now, for x-1, wr get, 

`f(x)=f(-1)=4(-1)^3+8(-1)^2+(-1)-3=-4+8-1-3=0` 

Hence, (x+1) is a factor of f (x) 

 

∴ `4x^3+8x^2+x-3=(x+1)(4x^2+4x-3) `

                           =`(x+1)(4x^2+6x-2x-3) `

                           =`(x+1)[2x(2x+3)-(2x+3)]`

                          =` (x+1)(2x+3)(2x-1) `

 

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Solution The expression 4x3 – bx2 + x – c leaves remainders 0 and 30 when divided by x + 1 and 2x – 3 respectively. Calculate the values of b and c. Hence, factorise the expression completely. Concept: Factorising a Polynomial Completely After Obtaining One Factor by Factor Theorem.
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