Question

$5x3 + 2x2 + x + 4 \div x + 2$
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Answer 1

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Answer 2

ꜱᴛᴇᴘ ʙʏ ꜱᴛᴇᴘ ꜱᴏʟᴜᴛɪᴏɴ :

ꜱᴛᴇᴘ 1 :

ꜱɪᴍᴘʟɪꜰʏ : 4/x

ᴇQᴜᴀᴛɪᴏɴ ᴀᴛ ᴛʜᴇ ᴇɴᴅ ᴏꜰ ꜱᴛᴇᴘ 1 :

( ( ( (5• ( x³) ) + (2• ( x ²) ) ) + x) + 4/x ) +2

ꜱᴛᴇᴘ 2 :

ᴇQᴜᴀᴛɪᴏɴ ᴀᴛ ᴛʜᴇ ᴇɴᴅ ᴏꜰ ꜱᴛᴇᴘ 2:

( ( ( (5• (x³) ) + 2 x ²) +x) + 4/x )+2

ꜱᴛᴇᴘ 3 :

ᴇQᴜᴀᴛɪᴏɴ ᴀᴛ ᴛʜᴇ ᴇɴᴅ ᴏꜰ ꜱᴛᴇᴘ 3:

( ( (5 x ³ + 2 x ³ ) + x) + 4/x ) + 2

ꜱᴛᴇᴘ 4 :

ʀᴇᴡʀɪᴛɪɴɢ ᴛʜᴇ ᴡʜᴏʟᴇ ᴀꜱ ᴀɴ ᴇQᴜɪᴠᴀʟᴇɴᴛ ꜰʀᴀᴄᴛɪᴏɴ :

4 . 1 ᴀᴅᴅɪɴɢ ᴀ ꜰʀᴀᴄᴛɪᴏɴ ᴛᴏ ᴀ ᴡʜᴏʟᴇ

ʀᴇᴡʀɪᴛᴇ ᴛʜᴇ ᴡʜᴏʟᴇ ᴀꜱ ᴀ ꜰʀᴀᴄᴛɪᴏɴ ᴜꜱɪɴɢ x ᴀꜱ ᴛʜᴇ ᴅᴇɴᴏᴍɪɴᴀᴛᴏʀ :

5x ³ + 2x ² + x = 5x ³ + 2x ² + x /1 = (5x ³+ 2x ² + x)•x/x

ᴇQᴜɪᴠᴀʟᴇɴᴛ ꜰʀᴀᴄᴛɪᴏɴ :

ᴛʜᴇ ꜰʀᴀᴄᴛɪᴏɴ ᴛʜᴜꜱ ɢᴇɴᴇʀᴀᴛᴇᴅ ʟᴏᴏᴋꜱ ᴅɪꜰꜰᴇʀᴇɴᴛ ʙᴜᴛ ʜᴀꜱ ᴛʜᴇ ꜱᴀᴍᴇ ᴠᴀʟᴜᴇ ᴀꜱ ᴛʜᴇ ᴡʜᴏʟᴇ

ᴄᴏᴍᴍᴏɴ ᴅᴇɴᴏᴍɪɴᴀᴛᴏʀ :

ᴛʜᴇ ᴇQᴜɪᴠᴀʟᴇɴᴛ ꜰʀᴀᴄᴛɪᴏɴ ᴀɴᴅ ᴛʜᴇ ᴏᴛʜᴇʀ ꜰʀᴀᴄᴛɪᴏɴ ɪɴᴠᴏʟᴠᴇᴅ ɪɴ ᴛʜᴇ ᴄᴀʟᴄᴜʟᴀᴛɪᴏɴ ꜱʜᴀʀᴇ ᴛʜᴇ ꜱᴀᴍᴇ ᴅᴇɴᴏᴍɪɴᴀᴛᴏʀ

ꜱᴛᴇᴘ 5 :

ᴘᴜʟʟɪɴɢ ᴏᴜᴛ ʟɪᴋᴇ ᴛᴇʀᴍꜱ

5 . 1 ᴘᴜʟʟ ᴏᴜᴛ ʟɪᴋᴇ ꜰᴀᴄᴛᴏʀꜱ :

5x ³ + 2x ³ + x = x • (5x ²+ 2x + 1)

ᴛʀʏɪɴɢ ᴛᴏ ꜰᴀᴄᴛᴏʀ ʙʏ ꜱᴘʟɪᴛᴛɪɴɢ ᴛʜᴇ ᴍɪᴅᴅʟᴇ ᴛᴇʀᴍ :

5 . 2 ꜰᴀᴄᴛᴏʀɪɴɢ 5x2 + 2x + 1

ᴛʜᴇ ꜰɪʀꜱᴛ ᴛᴇʀᴍ ɪꜱ, 5x² ɪᴛꜱ ᴄᴏᴇꜰꜰɪᴄɪᴇɴᴛ ɪꜱ 5 .

ᴛʜᴇ ᴍɪᴅᴅʟᴇ ᴛᴇʀᴍ ɪꜱ, +2x ɪᴛꜱ ᴄᴏᴇꜰꜰɪᴄɪᴇɴᴛ ɪꜱ 2 .

ᴛʜᴇ ʟᴀꜱᴛ ᴛᴇʀᴍ, "ᴛʜᴇ ᴄᴏɴꜱᴛᴀɴᴛ", ɪꜱ +1 .

ꜱᴛᴇᴘ-1 : ᴍᴜʟᴛɪᴘʟʏ ᴛʜᴇ ᴄᴏᴇꜰꜰɪᴄɪᴇɴᴛ ᴏꜰ ᴛʜᴇ ꜰɪʀꜱᴛ ᴛᴇʀᴍ ʙʏ ᴛʜᴇ ᴄᴏɴꜱᴛᴀɴᴛ 5 • 1 = 5

ꜱᴛᴇᴘ-2 : ꜰɪɴᴅ ᴛᴡᴏ ꜰᴀᴄᴛᴏʀꜱ ᴏꜰ 5 ᴡʜᴏꜱᴇ ꜱᴜᴍ ᴇQᴜᴀʟꜱ ᴛʜᴇ ᴄᴏᴇꜰꜰɪᴄɪᴇɴᴛ ᴏꜰ ᴛʜᴇ ᴍɪᴅᴅʟᴇ ᴛᴇʀᴍ, ᴡʜɪᴄʜ ɪꜱ 2 .

-5 + -1 = -6

-1 + -5 = -6

1 + 5 = 6

5 + 1 = 6

ᴏʙꜱᴇʀᴠᴀᴛɪᴏɴ :

ɴᴏ ᴛᴡᴏ ꜱᴜᴄʜ ꜰᴀᴄᴛᴏʀꜱ ᴄᴀɴ ʙᴇ ꜰᴏᴜɴᴅ !!

ᴄᴏɴᴄʟᴜꜱɪᴏɴ :

ᴛʀɪɴᴏᴍɪᴀʟ ᴄᴀɴ ɴᴏᴛ ʙᴇ ꜰᴀᴄᴛᴏʀᴇᴅ

ᴀᴅᴅɪɴɢ ꜰʀᴀᴄᴛɪᴏɴꜱ ᴛʜᴀᴛ ʜᴀᴠᴇ ᴀ ᴄᴏᴍᴍᴏɴ ᴅᴇɴᴏᴍɪɴᴀᴛᴏʀ :

5.3 ᴀᴅᴅɪɴɢ ᴜᴘ ᴛʜᴇ ᴛᴡᴏ ᴇQᴜɪᴠᴀʟᴇɴᴛ ꜰʀᴀᴄᴛɪᴏɴꜱ.

ᴀᴅᴅ ᴛʜᴇ ᴛᴡᴏ ᴇQᴜɪᴠᴀʟᴇɴᴛ ꜰʀᴀᴄᴛɪᴏɴꜱ ᴡʜɪᴄʜ ɴᴏᴡ ʜᴀᴠᴇ ᴀ ᴄᴏᴍᴍᴏɴ ᴅᴇɴᴏᴍɪɴᴀᴛᴏʀ

ᴄᴏᴍʙɪɴᴇ ᴛʜᴇ ɴᴜᴍᴇʀᴀᴛᴏʀꜱ ᴛᴏɢᴇᴛʜᴇʀ, ᴘᴜᴛ ᴛʜᴇ ꜱᴜᴍ ᴏʀ ᴅɪꜰꜰᴇʀᴇɴᴄᴇ ᴏᴠᴇʀ ᴛʜᴇ ᴄᴏᴍᴍᴏɴ ᴅᴇɴᴏᴍɪɴᴀᴛᴏʀ ᴛʜᴇɴ ʀᴇᴅᴜᴄᴇ ᴛᴏ ʟᴏᴡᴇꜱᴛ ᴛᴇʀᴍꜱ ɪꜰ ᴘᴏꜱꜱɪʙʟᴇ:

x • (5x²+2x+1) • x + 4/x = 5x⁴ + 2x³ + x² + 4 /x

ᴇQᴜᴀᴛɪᴏɴ ᴀᴛ ᴛʜᴇ ᴇɴᴅ ᴏꜰ ꜱᴛᴇᴘ 5 :

(5x⁴ + 2x³ + x² + 4)/x + 2

ꜱᴛᴇᴘ 6 :

ʀᴇᴡʀɪᴛɪɴɢ ᴛʜᴇ ᴡʜᴏʟᴇ ᴀꜱ ᴀɴ ᴇQᴜɪᴠᴀʟᴇɴᴛ ꜰʀᴀᴄᴛɪᴏɴ :

6.1 ᴀᴅᴅɪɴɢ ᴀ ᴡʜᴏʟᴇ ᴛᴏ ᴀ ꜰʀᴀᴄᴛɪᴏɴ

ʀᴇᴡʀɪᴛᴇ ᴛʜᴇ ᴡʜᴏʟᴇ ᴀꜱ ᴀ ꜰʀᴀᴄᴛɪᴏɴ ᴜꜱɪɴɢ x ᴀꜱ ᴛʜᴇ ᴅᴇɴᴏᴍɪɴᴀᴛᴏʀ :

2 = 2/1 = 2 • x /x

ᴄʜᴇᴄᴋɪɴɢ ꜰᴏʀ ᴀ ᴘᴇʀꜰᴇᴄᴛ ᴄᴜʙᴇ :

6.2 5x⁴ + 2x³ + x² + 4 ɪꜱ ɴᴏᴛ ᴀ ᴘᴇʀꜰᴇᴄᴛ ᴄᴜʙᴇ

ᴛʀʏɪɴɢ ᴛᴏ ꜰᴀᴄᴛᴏʀ ʙʏ ᴘᴜʟʟɪɴɢ ᴏᴜᴛ :

6.3 ꜰᴀᴄᴛᴏʀɪɴɢ:

5x⁴ + 2x³ + x² + 4

ᴛʜᴏᴜɢʜᴛꜰᴜʟʟʏ ꜱᴘʟɪᴛ ᴛʜᴇ ᴇxᴘʀᴇꜱꜱɪᴏɴ ᴀᴛ ʜᴀɴᴅ ɪɴᴛᴏ ɢʀᴏᴜᴘꜱ, ᴇᴀᴄʜ ɢʀᴏᴜᴘ ʜᴀᴠɪɴɢ ᴛᴡᴏ ᴛᴇʀᴍꜱ :

• ɢʀᴏᴜᴘ 1: x2 + 4
• ɢʀᴏᴜᴘ 2: 5x4 + 2x3

ᴘᴜʟʟ ᴏᴜᴛ ꜰʀᴏᴍ ᴇᴀᴄʜ ɢʀᴏᴜᴘ ꜱᴇᴘᴀʀᴀᴛᴇʟʏ :

• ɢʀᴏᴜᴘ 1: (x2 + 4) • (1)
• ɢʀᴏᴜᴘ 2: (5x + 2) • (x3)

ʙᴀᴅ ɴᴇᴡꜱ !! ꜰᴀᴄᴛᴏʀɪɴɢ ʙʏ ᴘᴜʟʟɪɴɢ ᴏᴜᴛ ꜰᴀɪʟꜱ :

ᴛʜᴇ ɢʀᴏᴜᴘꜱ ʜᴀᴠᴇ ɴᴏ ᴄᴏᴍᴍᴏɴ ꜰᴀᴄᴛᴏʀ ᴀɴᴅ ᴄᴀɴ ɴᴏᴛ ʙᴇ ᴀᴅᴅᴇᴅ ᴜᴘ ᴛᴏ ꜰᴏʀᴍ ᴀ ᴍᴜʟᴛɪᴘʟɪᴄᴀᴛɪᴏɴ.

ᴘᴏʟʏɴᴏᴍɪᴀʟ ʀᴏᴏᴛꜱ ᴄᴀʟᴄᴜʟᴀᴛᴏʀ :

6.4 ꜰɪɴᴅ ʀᴏᴏᴛꜱ (ᴢᴇʀᴏᴇꜱ) ᴏꜰ :

ꜰ(x) = 5x⁴ + 2x³ + x² + 4

ᴘᴏʟʏɴᴏᴍɪᴀʟ ʀᴏᴏᴛꜱ ᴄᴀʟᴄᴜʟᴀᴛᴏʀ ɪꜱ ᴀ ꜱᴇᴛ ᴏꜰ ᴍᴇᴛʜᴏᴅꜱ ᴀɪᴍᴇᴅ ᴀᴛ ꜰɪɴᴅɪɴɢ ᴠᴀʟᴜᴇꜱ ᴏꜰ x ꜰᴏʀ ᴡʜɪᴄʜ ꜰ(x)=0

ʀᴀᴛɪᴏɴᴀʟ ʀᴏᴏᴛꜱ ᴛᴇꜱᴛ ɪꜱ ᴏɴᴇ ᴏꜰ ᴛʜᴇ ᴀʙᴏᴠᴇ ᴍᴇɴᴛɪᴏɴᴇᴅ ᴛᴏᴏʟꜱ. ɪᴛ ᴡᴏᴜʟᴅ ᴏɴʟʏ ꜰɪɴᴅ ʀᴀᴛɪᴏɴᴀʟ ʀᴏᴏᴛꜱ ᴛʜᴀᴛ ɪꜱ ɴᴜᴍʙᴇʀꜱ x ᴡʜɪᴄʜ ᴄᴀɴ ʙᴇ ᴇxᴘʀᴇꜱꜱᴇᴅ ᴀꜱ ᴛʜᴇ Qᴜᴏᴛɪᴇɴᴛ ᴏꜰ ᴛᴡᴏ ɪɴᴛᴇɢᴇʀꜱ

ᴛʜᴇ ʀᴀᴛɪᴏɴᴀʟ ʀᴏᴏᴛ ᴛʜᴇᴏʀᴇᴍ ꜱᴛᴀᴛᴇꜱ ᴛʜᴀᴛ ɪꜰ ᴀ ᴘᴏʟʏɴᴏᴍɪᴀʟ ᴢᴇʀᴏᴇꜱ ꜰᴏʀ ᴀ ʀᴀᴛɪᴏɴᴀʟ ɴᴜᴍʙᴇʀ ᴘ/Q ᴛʜᴇɴ ᴘ ɪꜱ ᴀ ꜰᴀᴄᴛᴏʀ ᴏꜰ ᴛʜᴇ ᴛʀᴀɪʟɪɴɢ ᴄᴏɴꜱᴛᴀɴᴛ ᴀɴᴅ Q ɪꜱ ᴀ ꜰᴀᴄᴛᴏʀ ᴏꜰ ᴛʜᴇ ʟᴇᴀᴅɪɴɢ ᴄᴏᴇꜰꜰɪᴄɪᴇɴᴛ

ɪɴ ᴛʜɪꜱ ᴄᴀꜱᴇ, ᴛʜᴇ ʟᴇᴀᴅɪɴɢ ᴄᴏᴇꜰꜰɪᴄɪᴇɴᴛ ɪꜱ 5 ᴀɴᴅ ᴛʜᴇ ᴛʀᴀɪʟɪɴɢ ᴄᴏɴꜱᴛᴀɴᴛ ɪꜱ 4.

ꜰɪɴᴀʟ ʀᴇꜱᴜʟᴛ :

5x⁴ + 2x³ + x² + 2x + 4/x