Question

(x2-x-2) is factor of x3 + 3x2 + ax + b, calculate the values of a and b. Write all
the factors of the given expression.​

Answer 1

SOLUTION

GIVEN

$\sf{ ({x}^{2} - x - 2) \: \: is \: a \: factor \: of \: {x}^{3} + 3 {x}^{2} + ax + b }$

TO DETERMINE

• The values of a and b

• All factors of the given expression

EVALUATION

Here it is given that

$\sf{ ({x}^{2} - x - 2) \: \: is \: a \: factor \: of \: {x}^{3} + 3 {x}^{2} + ax + b }$

Now

$\sf{ ({x}^{2} - x - 2) \: }$

$\sf{ = ({x}^{2} - 2x + x- 2) \: }$

$\sf{ = x(x - 2) + 1(x - 2) }$

$\sf{ = (x - 2) (x + 1) }$

Hence the zeroes are - 1 & 2

Since 2 is a zero of the given expression

8 + 12 + 2a + b = 0

$\implies \sf{2a + b = - 20} \: \: \: - - - (1)$

Again - 1 is a zero of the expression

- 1 + 3 - a + b = 0

$\implies \sf{a - b = 2} \: \: \: - - - (2)$

Equation 1 + Equation 2 gives

3a = - 18

∴ a = - 6

∴ b = - 8

Hence the required values are

a = - 6 & b = - 8

Hence the given expression becomes

$\sf{{x}^{3} + 3 {x}^{2} - 6x - 8 }$

$= \sf{x( {x}^{2} - x - 2) + 4( {x}^{2} - x - 2) }$

$= \sf{(x + 4)( {x}^{2} - x - 2) }$

$= \sf{(x + 4)(x - 2)(x + 1) }$

Hence all three factors are

(x+4) , (x-2) , (x+1)

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Answer 2

Solution ⤵️

Given ⤵️

$\sf{ ({x}^{2} - x - 2) \: \: is \: a \: factor \: of \: {x}^{3} + 3 {x}^{2} + ax + b }$

To Find ⤵️

• The values of a and b

• All factors of the given expression

Calculation ⤵️

Here it is given that,

$\sf{ ({x}^{2} - x - 2) \: \: is \: a \: factor \: of \: {x}^{3} + 3 {x}^{2} + ax + b }$

Now,

$\sf{ ({x}^{2} - x - 2) \: }$

$\sf{ = ({x}^{2} - 2x + x- 2) \: }$

$\sf{ = x(x - 2) + 1(x - 2) }$

$\sf{ = (x - 2) (x + 1) }$

Hence the zeroes are - 1 and 2

Since, 2 is a zero of the given expression,

8 + 12 + 2a + b = 0

$\implies \sf{2a + b = - 20} \: \: \: - - - (1)$

Again, - 1 is a zero of the expression,

- 1 + 3 - a + b = 0

$\implies \sf{a - b = 2} \: \: \: - - - (2)$

Equation (1) + Equation (2) gives,

3a = - 18

∴ a = - 6

∴ b = - 8

Hence the required values are,

a = - 6 & b = - 8

Hence the given expression becomes,

$\sf{{x}^{3} + 3 {x}^{2} - 6x - 8 }$

$= \sf{x( {x}^{2} - x - 2) + 4( {x}^{2} - x - 2) }$

$= \sf{(x + 4)( {x}^{2} - x - 2) }$

$= \sf{(x + 4)(x - 2)(x + 1) }$

Hence all three factors are,

(x+4) , (x-2) , (x+1)

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