Question

7. p (x) is a polynomial of degree more than 2. When p (x) is divided by x - 2, it leaves remainder 1 and when it is divided by x - 3 it leaves a remainder 3. Find the remainder when p ( x ) is divided by ( x - 2 )( x - 3 ).

Standard:- 10

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Answer 1

Given that when p(x) is divided by x - 2, it leaves remainder 1.

p(x) = Q1(x - 2) + 1

p(2) = 1

Given that when p(x) is divided by x - 3, it leaves remainder 3.

p(x) = Q2(x - 3) + 3

p(3) = 3.

Now,

Let the remainder be ax + b.

p(x) = Q3(x - 2)(x - 3) + ax + b

Substitute x = 2 and x = 3, we get

1 = 2a + b   ------- (1)

3 = 3a + b    ------- (2)


On solving (1) & (2), we get

2a + b = 1

3a + b = 3

-------------------

-a = -2

a = 2

Substitute a = 2 in (1), we get

2a + b = 1

2(2) + b = 1

4 + b = 1

b = -3.

Hence, ax + b = 2x - 3.


Therefore, when p(x) is divided by (x - 2)(x - 3) it leaves remainder (2x - 3).


Hope it helps!

Answer 2

ɢɪᴠᴇɴ ᴛʜᴀᴛ ᴡʜᴇɴ ᴘ(x) ɪs ᴅɪᴠɪᴅᴇᴅ ʙʏ x - 2, ɪᴛ ʟᴇᴀᴠᴇs ʀᴇᴍᴀɪɴᴅᴇʀ 1.

ᴘ(x) = ϙ1(x - 2) + 1

ᴘ(2) = 1

ɢɪᴠᴇɴ ᴛʜᴀᴛ ᴡʜᴇɴ ᴘ(x) ɪs ᴅɪᴠɪᴅᴇᴅ ʙʏ x - 3, ɪᴛ ʟᴇᴀᴠᴇs ʀᴇᴍᴀɪɴᴅᴇʀ 3.

ᴘ(x) = ϙ2(x - 3) + 3

ᴘ(3) = 3.

ɴᴏᴡ,

ʟᴇᴛ ᴛʜᴇ ʀᴇᴍᴀɪɴᴅᴇʀ ʙᴇ ᴀx + ʙ.

ᴘ(x) = ϙ3(x - 2)(x - 3) + ᴀx + ʙ

sᴜʙsᴛɪᴛᴜᴛᴇ x = 2 ᴀɴᴅ x = 3, ᴡᴇ ɢᴇᴛ

1 = 2ᴀ + ʙ   ------- (1)

3 = 3ᴀ + ʙ    ------- (2)

ᴏɴ sᴏʟᴠɪɴɢ (1) & (2), ᴡᴇ ɢᴇᴛ

2ᴀ + ʙ = 1

3ᴀ + ʙ = 3

-------------------

-ᴀ = -2

ᴀ = 2

sᴜʙsᴛɪᴛᴜᴛᴇ ᴀ = 2 ɪɴ (1), ᴡᴇ ɢᴇᴛ

2ᴀ + ʙ = 1

2(2) + ʙ = 1

4 + ʙ = 1

ʙ = -3.

ʜᴇɴᴄᴇ, ᴀx + ʙ = 2x - 3.

ᴛʜᴇʀᴇғᴏʀᴇ, ᴡʜᴇɴ ᴘ(x) ɪs ᴅɪᴠɪᴅᴇᴅ ʙʏ (x - 2)(x - 3) ɪᴛ ʟᴇᴀᴠᴇs ʀᴇᴍᴀɪɴᴅᴇʀ (2x - 3).