Question

If (x + ax + 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.​

Answer 1

sᴏʟᴜᴛɪᴏɴ

ʟᴇғᴛ ғ(x) =

x3+ᴀx2+ʙx+6

x-2=0⇒ x=2

sɪɴᴄᴇ, x-2 ɪs ᴀ ғᴀᴄᴛᴏʀ, ʀᴇᴍᴀɪɴᴅᴇʀ=0

∴ ғ(2)=0

(2)3+ᴀ(2)2+ʙ(2)+6=0

8+4ᴀ+2ʙ+6=0

2ᴀ+ʙ+7=0 .........(1)

ᴏɴ ᴅɪᴠɪᴅɪɴɢ ғ(x)ʙʏ x-3, ɪᴛ ʟᴇᴀᴠᴇs ᴀ ʀᴇᴍᴀɪɴᴅᴇʀ 3.

∴ ғ(3)=3

(3)3+ᴀ(3)2+ʙ(3)+6=3

27+9ᴀ+3ʙ+6=3

3ᴀ+ʙ+10 ...........(2)

sᴜʙᴛʀᴀᴄᴛɪɴɢ (1) ғʀᴏᴍ (2), ᴡᴇ ɢᴇᴛ,

ᴀ+3=0

ᴀ=-3

sᴜʙsᴛɪᴛᴜᴛɪɴɢ ᴛʜᴇ ᴠᴀʟᴜᴇ ᴏғ ᴀ ɪɴ (1), ᴡᴇ ɢᴇᴛ,

-6+ʙ+7=0

ʙ=-1