Question

If the polynomial t³ - 3t² + kt + 50 is divided by (t-3) , the remainder is 62. Find the value of K.

Answer 1

$\huge\bf{\underline{\underline{Given -}}}$

★ Wʜᴇɴ Gɪᴠᴇɴ Pᴏʟʏɴᴏᴍɪᴀʟ Iꜱ Dɪᴠɪᴅᴇᴅ Bʏ (ᴛ-3) Tʜᴇ Rᴇᴍᴀɪɴᴅᴇʀ ɪꜱ 62 . Iᴛ Mᴇᴀɴꜱ ᴛʜᴇ Vᴀʟᴜᴇ Oꜰ ᴛʜᴇ Pᴏʟʏɴᴏᴍɪᴀʟ Wʜᴇɴ ᴛ = 3 ɪꜱ 62.

$p(t) = {t}^{3} - 3 {t}^{2} + kt + 50$

$\huge\bf{\underline{\underline{SolutiOn -}}}$

★ Bʏ Rᴇᴍᴀɪɴᴅᴇʀ Tʜᴇᴏʀᴇᴍ ,

Rᴇᴍᴀɪɴᴅᴇʀ =

$p(3) \: = {3}^{3} - 3 \times {3}^{2} + k \times 3 + 50 \\ \: \: \: = 27 - 3 \times 9 + 3k + 50 \\ = 27 - 27 + 3k + 50 \: \: \\ = 3k + 50 \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \:$

★ Bᴜᴛ Rᴇᴍᴀɪɴᴅᴇʀ Iꜱ 62 .

Sᴏ,

$\therefore \: 3k + 50 = 62 \\ \therefore3k = 62 - 50 \\ \therefore3k = 12 \: \: \: \: \: \: \: \: \: \\ \therefore \: k = \frac{12}{3} \: \: \: \: \: \: \: \: \: \\ \therefore \: k = 4 \: \: \: \: \: \: \: \: \: \: \:$

★ Sᴏ ᴛʜᴇ Vᴀʟᴜᴇ Oꜰ $\fbox{k = 4}$

Answer 2

Answer :

›»› The value of k is 4.

Given :

• The polynomial t³ - 3t² + kt + 50 is divided by (t - 3), the remainder is 62.

To Find :

• The value of k = ?

Solution :

When the polynomial t³ - 3t² + kt + 50 is divided by (t-3) , the remainder is 62. This means the value of polynomial when t = 3 is 62.

• p(t) = t³ - 3t² + kt + 50

From remainder theorem

→ Reminder = p(3) = 3³ - 3 * 3² + k * 3 + 50

→ Reminder = p(3) = 27 - 3 * 9 + k * 3 + 50

→ Reminder = p(3) = 27 - 27 + 3k + 50

→ Reminder = p(3) = 0 + 3k + 50

→ Reminder = p(3) = 3k + 50

But remainder 62 so,

→ 3k + 50 = 62

→ 3k = 62 - 50

→ 3k = 12

→ k = 4

║Hence, the value of k is 4.║