Question

Question:

For a polynomial p(x), the value of p(3) is −2. Which of the following must be true about p(x)?

A) x−5 is a factor of p(x).
B) x−2 is a factor of p(x).
C) x+2 is a factor of p(x).
D) The remainder when p(x) is divided by x−3 is −2.​

Answer 1

The only one that must be true about p(x) is D, that the remainder when p(x) is divided by x−3 is -2.

The final answer is D.

Answer 2

Answer:

ᴀɴꜱᴡᴇʀ ᴇxᴘʟᴀɴᴀᴛɪᴏɴ:

ɪꜰ ᴛʜᴇ ᴘᴏʟʏɴᴏᴍɪᴀʟ ᴘ(x) ɪꜱ ᴅɪᴠɪᴅᴇᴅ ʙʏ ᴀ ᴘᴏʟʏɴᴏᴍɪᴀʟ ᴏꜰ ᴛʜᴇ ꜰᴏʀᴍ x+ᴋ (ᴡʜɪᴄʜ ᴀᴄᴄᴏᴜɴᴛꜱ ꜰᴏʀ ᴀʟʟ ᴏꜰ ᴛʜᴇ ᴘᴏꜱꜱɪʙʟᴇ ᴀɴꜱᴡᴇʀ ᴄʜᴏɪᴄᴇꜱ ɪɴ ᴛʜɪꜱ Qᴜᴇꜱᴛɪᴏɴ), ᴛʜᴇ ʀᴇꜱᴜʟᴛ ᴄᴀɴ ʙᴇ ᴡʀɪᴛᴛᴇɴ ᴀꜱ

ᴘ(x)/x+ᴋ = Q(x) + ʀ/x + ᴋ

ᴡʜᴇʀᴇ Q(x) ɪꜱ ᴀ ᴘᴏʟʏɴᴏᴍɪᴀʟ ᴀɴᴅ ʀ ɪꜱ ᴛʜᴇ ʀᴇᴍᴀɪɴᴅᴇʀ. ꜱɪɴᴄᴇ x+ᴋ ɪꜱ ᴀ ᴅᴇɢʀᴇᴇ-1 ᴘᴏʟʏɴᴏᴍɪᴀʟ (ᴍᴇᴀɴɪɴɢ ɪᴛ ᴏɴʟʏ ɪɴᴄʟᴜᴅᴇꜱ x1 ᴀɴᴅ ɴᴏ ʜɪɢʜᴇʀ ᴇxᴘᴏɴᴇɴᴛꜱ), ᴛʜᴇ ʀᴇᴍᴀɪɴᴅᴇʀ ɪꜱ ᴀ ʀᴇᴀʟ ɴᴜᴍʙᴇʀ.

ᴛʜᴇʀᴇꜰᴏʀᴇ, ᴘ(x) ᴄᴀɴ ʙᴇ ʀᴇᴡʀɪᴛᴛᴇɴ ᴀꜱ ᴘ(x)=(x+ᴋ)Q(x)+ʀ, ᴡʜᴇʀᴇ ʀ ɪꜱ ᴀ ʀᴇᴀʟ ɴᴜᴍʙᴇʀ.

ᴛʜᴇ Qᴜᴇꜱᴛɪᴏɴ ꜱᴛᴀᴛᴇꜱ ᴛʜᴀᴛ ᴘ(3)=−2, ꜱᴏ ɪᴛ ᴍᴜꜱᴛ ʙᴇ ᴛʀᴜᴇ ᴛʜᴀᴛ

−2=ᴘ(3)=(3+ᴋ)Q(3)+ʀ

ɴᴏᴡ ᴡᴇ ᴄᴀɴ ᴘʟᴜɢ ɪɴ ᴀʟʟ ᴛʜᴇ ᴘᴏꜱꜱɪʙʟᴇ ᴀɴꜱᴡᴇʀꜱ. ɪꜰ ᴛʜᴇ ᴀɴꜱᴡᴇʀ ɪꜱ ᴀ, ʙ, ᴏʀ ᴄ, ʀ ᴡɪʟʟ ʙᴇ 0, ᴡʜɪʟᴇ ɪꜰ ᴛʜᴇ ᴀɴꜱᴡᴇʀ ɪꜱ ᴅ, ʀ ᴡɪʟʟ ʙᴇ −2.

ᴀ. −2=ᴘ(3)=(3+(−5))Q(3)+0

−2=(3−5)Q(3)

−2=(−2)Q(3)

ᴛʜɪꜱ ᴄᴏᴜʟᴅ ʙᴇ ᴛʀᴜᴇ, ʙᴜᴛ ᴏɴʟʏ ɪꜰ Q(3)=1

ʙ. −2=ᴘ(3)=(3+(−2))Q(3)+0

−2=(3−2)Q(3)

−2=(−1)Q(3)

ᴛʜɪꜱ ᴄᴏᴜʟᴅ ʙᴇ ᴛʀᴜᴇ, ʙᴜᴛ ᴏɴʟʏ ɪꜰ Q(3)=2

ᴄ. −2=ᴘ(3)=(3+2)Q(3)+0

−2=(5)Q(3)

ᴛʜɪꜱ ᴄᴏᴜʟᴅ ʙᴇ ᴛʀᴜᴇ, ʙᴜᴛ ᴏɴʟʏ ɪꜰ Q(3) = −2/5

ᴅ. −2=ᴘ(3)=(3+(−3))Q(3)+(−2)

−2=(3−3)Q(3)+(−2)

−2=(0)Q(3)+(−2)

ᴛʜɪꜱ ᴡɪʟʟ ᴀʟᴡᴀʏꜱ ʙᴇ ᴛʀᴜᴇ ɴᴏ ᴍᴀᴛᴛᴇʀ ᴡʜᴀᴛ Q(3) ɪꜱ.

ᴏꜰ ᴛʜᴇ ᴀɴꜱᴡᴇʀ ᴄʜᴏɪᴄᴇꜱ, ᴛʜᴇ ᴏɴʟʏ ᴏɴᴇ ᴛʜᴀᴛ ᴍᴜꜱᴛ ʙᴇ ᴛʀᴜᴇ ᴀʙᴏᴜᴛ ᴘ(x) ɪꜱ ᴅ, ᴛʜᴀᴛ ᴛʜᴇ ʀᴇᴍᴀɪɴᴅᴇʀ ᴡʜᴇɴ ᴘ(x) ɪꜱ ᴅɪᴠɪᴅᴇᴅ ʙʏ x−3 ɪꜱ -2.

ᴛʜᴇ ꜰɪɴᴀʟ ᴀɴꜱᴡᴇʀ ɪꜱ ᴅ.