P(x)=2x²-5x-6,having zeros as alpha and beta,

to find -->f(x) such that its zeros are (alpha +beta)and alpha×beta,

in p(x) sum of zeros=alpha+beta=-b/a,

alpha+beta=-(-5)/2=5/2,

product of zeros=alpha×beta=c/a,

alpha×beta=-6/2=-3,

zeros of f(x) are 5/2 and -3,

f(x)=x²-( sum of zeros)x+ product of zeros,

=x²-(5/2-3)x+(5/2×-3),

=x²-(-1/2)x-(15/2),

=x²+1/2x-15/2,

=2x²+x-15

Step-by-step explanation:

ᴘ(x)=2x²-5x-6,ʜᴀᴠɪɴɢ ᴢᴇʀᴏꜱ ᴀꜱ ᴀʟᴘʜᴀ ᴀɴᴅ ʙᴇᴛᴀ,

ᴛᴏ ꜰɪɴᴅ -->ꜰ(x) ꜱᴜᴄʜ ᴛʜᴀᴛ ɪᴛꜱ ᴢᴇʀᴏꜱ ᴀʀᴇ (ᴀʟᴘʜᴀ +ʙᴇᴛᴀ)ᴀɴᴅ ᴀʟᴘʜᴀ×ʙᴇᴛᴀ,

ɪɴ ᴘ(x) ꜱᴜᴍ ᴏꜰ ᴢᴇʀᴏꜱ=ᴀʟᴘʜᴀ+ʙᴇᴛᴀ=-ʙ/ᴀ,

ᴀʟᴘʜᴀ+ʙᴇᴛᴀ=-(-5)/2=5/2,

ᴘʀᴏᴅᴜᴄᴛ ᴏꜰ ᴢᴇʀᴏꜱ=ᴀʟᴘʜᴀ×ʙᴇᴛᴀ=ᴄ/ᴀ,

ᴀʟᴘʜᴀ×ʙᴇᴛᴀ=-6/2=-3,

ᴢᴇʀᴏꜱ ᴏꜰ ꜰ(x) ᴀʀᴇ 5/2 ᴀɴᴅ -3,

ꜰ(x)=x²-( ꜱᴜᴍ ᴏꜰ ᴢᴇʀᴏꜱ)x+ ᴘʀᴏᴅᴜᴄᴛ ᴏꜰ ᴢᴇʀᴏꜱ,

=x²-(5/2-3)x+(5/2×-3),

=x²-(-1/2)x-(15/2),

=x²+1/2x-15/2,

=2x²+x-15...