Question

If (2+x) and (3+x) are the factors of x³+ax+b,, find the value of a and b

Answer 1

$\textbf{ Hello Mate! }$

Since, ( x + 2 ) and ( x + 3 ) are factors then final value of p(x) = 0 where x = - 2 and - 3

$p(x) = {x}^{3} + ax + b \\ p( - 2) = { - 2}^{3} + a( - 2) + b \\0 = - 8 - 2a + b \\ 8 = - 2a + b .....(1)\\ \\ p( - 3) = { - 3}^{3} + a( - 3) + b \\ 0 = - 27 - 3a + b \\ 27 = - 3a + b \\ - 27 = - ( - 3a + b) \\ - 27 = 3a - b........(2)$

On adding (1) and (2) we get,

- 19 = a

Substituting values we get,

8 = - 2a + b

8 = - 2( - 19 ) + b

8 = 38 + b

- 30 = b

$\textsf{ \red{ a = - 19 and b = - 30 }}$

Have great future ahead!

Answer 2

Let p(x) be given Equation :-

${x}^{3} + ax + b$

→ Given (2+X) and (3+X) are the factors of given Equation

(1) → 2+ x = 0 ( since it's a factor )

→ X = -2

(2) → 3+ x = 0

→ X = -3

let's substitute X = -2 in given Equation

${ - 2}^{3} + a( - 2) + b \\ \\ - 8 - 2a + b \\ \\ - 2a + b = 8.............(1)$
let's substitute X= -3 in given Equation

${ - 3}^{3} + a( - 3) + b \\ \\ - 27 - 3a + b \\ \\ - 3a + b = 27..................(2)$

By Elimination Method

-3(-2a+b =8)

-2(-3a+b=27)

6a-3b = -24
6a-2b = -54
(-). (+). (+)
__________
-b = 30
__________

*************

b = -30

***************

substitute 'b' value in equation (1)

-2a + b = 8

-2a -30 = 8

-2a = 8+30

-2a = 38

a = 38/-2

*************

a = -19

**************

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VERIFICATION

${x}^{3} + ax + b \\ \\ { - 2}^{3} - 19( - 2) - 30 \\ \\ - 8 + 38 - 30 = 0$

hence [ a = -19 , b = -30 ]

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Thnq