Answer:
Hello,
Step-by-step explanation:
P(x)=x^3-5x^2-7x+51
Since the coefficients are all reals,
4+i (conjugate of 4-i) is also a root.
The polynomial est divisible by (x-4-i)(x-4+i)=(x-4)²+1=x²-8x+17
If we divide P(x) by x²-8x+17 we find the quotient (x+3) and the remainder 0
P(x)=(x+4)(x-4-i(x-4+i)
Roots are -4,4+i and 4-i
Answer:
All the roots are -3, 4-i and 4+i.
Step-by-step explanation:
If oine root is 4 - i then another one is 4 + i as complex roots occur as conjugate pairs.
(4 - i)(4 + i)
= 16 - i^2
= 17.
As the last term = 51 = 3 * 17
looks like the other root is 3 or -3.
By the Factor theorem
If x = 3 then f(3) = 0
f(3) = 27 - 5(3)^2 - 7(3) + 51
= 27 - 45 - 21 + 51 = 12 so 3 is not a root.
If x = -3:
f(-3) = -27 - 45 _ 21 + 51
= 0
So, x = -3 is a root.
