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If X3 + Ax2 + Bx + 6 Has X – 2 as a Factor and Leaves a Remainder 3 When Divided by X – 3, Find the Values of a and B. - ICSE Class 10 - Mathematics

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Question

If x3 + ax2 + bx + 6 has x – 2 as a factor and leaves a remainder 3 when divided by x – 3, find the values of a and b.

Solution

Leftf(x)=`x^3+ax^2+bx+6`   

`x-2=0`⇒ x=2

Since, x-2 is a factor, remainder=0 

∴ f(2)=0  

`(2)3+a(2)^2+b(2)+6=0`  

`8+4a+2b+6=0`   

`2a+b+7=0`  .........(1) 

On dividing f(x)by x-3, it leavvves a  remainder 3. 

∴ f(3)=3 

`(3)^3+a(3)^2+b(3)+6=3` 

`27+9a+3b+6=3` 

`3a+b+10`  ...........(2)

Subtracting (1) from (2), we get,  

a+3=0   

a=-3 

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

-6+b+7=0 

b=-1

 

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Solution If X3 + Ax2 + Bx + 6 Has X – 2 as a Factor and Leaves a Remainder 3 When Divided by X – 3, Find the Values of a and B. Concept: Remainder Theorem.
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