Question

If (x2-1) is a factor of px4+ qx3 + rx2 + sx + t, show that p + r + t = q + s = 0 .

Answer 1

ʜᴇʟʟᴏ ᴍᴀᴛᴇ!

x² - 1
ғᴀᴄᴛᴏʀ
x² - 1² = ( x + 1 )( x - 1 )

ɴᴏᴡ, x = - 1, + 1

ᴘ(1) + ǫ(1)³ + ʀ(1)² + s(1) + ᴛ
ᴘ + ǫ + ʀ + s + ᴛ = 0

ᴘ(-1)⁴ + ǫ(-1)³ + ʀ(-1)² + s(-1) + ᴛ
ᴘ - ǫ + ʀ - s + ᴛ

ᴏɴ ᴀᴅᴅɪɴɢ ʙᴏᴛʜ ᴇǫᴜᴀᴛɪᴏɴs ᴡᴇ ɢᴇᴛ

2ᴘ + 2ʀ + 2ᴛ ᴏʀ 2( ᴘ + ʀ + ᴛ ) = 0
ᴘ + ʀ + ᴛ = 0

ᴏɴ sᴜʙᴛʀᴀᴄᴛɪɴɢ ʙᴏᴛɢ ᴇǫᴜᴀᴛɪᴏɴᴇs ᴡᴇ ɢᴇᴛ

2 ( ǫ + s ) = 0
ǫ + s = 0

ʀʜs ɪs ᴇǫᴜᴀʟ ɪɴ ʙᴏᴛʜ ᴏᴜᴛᴄᴏᴍᴇs

ᴘ + ʀ + ᴛ = ǫ + s = 0