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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

ConceptFactorising a Polynomial Completely After Obtaining One Factor by Factor Theorem

#### 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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#### APPEARS IN

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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