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Given that X – 2 and X + 1 Are Factors of F(X) = X3 + 3x2 + Ax + B; Calculate the Values of a and B. Hence, Find All the Factors of F(X). - ICSE Class 10 - Mathematics

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Question

Given that x – 2 and x + 1 are factors of f(x) = x3 + 3x2 + ax + b; calculate the values of a and b. Hence, find all the factors of f(x).

Solution

`f(x)=x^3+3x^2+ax+b `

`"Since", (x-2)  "is a factor of"  f(x),f(2) =0` 

⇒ `(2)^3+3(2)^2 +a(-1)+b=0` 

⇒ `8+12+2a+b=0` 

 ⇒ `2a+b+20=0 `...........(1) 

Since, `(x+1)  "is a factor of"  f(x), f(-1)=0` 

`(-1)^3+3(-1)^2+a(-1)+b=0` 

`-1+3-a+b=0 `

`-a+b+2=0 `.............(2) 

Subtracting (2) from (1) we get, 

3a+18=0 

 ⇒ a=-6 

Subtracting the value of a in (2), we get, 

`b=a-2=-6-2=-8 ` 

∴ ` f(x)=x^3+3x^2-6x-8` 

`"Now, for x"=-1` 

`f(x)=f(-1)=(-1)^3+3(-1)^2-6(-1)-8 =-1+3+6-8=0` 

Hence, (x+1) is a factor of f (x) 

  

∴ `x^3+3x^2-6x-8=(x-1)(x^2+2x-8)` 

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

                             =`(x+1)[] x(x+4)-2(x+4)` 

                             =`(x+1)(x+4)(x-2)`

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Solution Given that X – 2 and X + 1 Are Factors of F(X) = X3 + 3x2 + Ax + B; Calculate the Values of a and B. Hence, Find All the Factors of F(X). Concept: Factorising a Polynomial Completely After Obtaining One Factor by Factor Theorem.
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