What number must be added to the expression below to complete the square?
1 answer:
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Answer:
f(x) = - 4x³ + 14x² - 36x + 45
Step-by-step explanation:
Given zeros x = a, x = b, then the factors are
(x - a) and (x - b)
and the polynomial is the product of the factors
Note that complex zeros occur in conjugate pairs
3i is a zero then - 3i is a zero
2 - i is a zero then 2 + i is a zero
The corresponding factors are
(x - 3i), (x - (- 3i), (x - (2 - i)) and (x - (2 + i)), that is
(x - 3i), (x + 3i), ((x - 2) + i), ((x - 2) - i)
Thus the polynomial is
f(x) = (x - 3i)(x + 3i)((x - 2) + i )((x - 2) - i) ← expand factors in pairs
= (x² - 9i² )((x - 2)² - i² ) → i² = - 1
= (x² + 9)(x² - 4x + 4 + 1)
= (x² + 9)(x² - 4x + 5) ← distribute
= - 4x³ + 5x² + 9x² - 36x + 45 ← collect like terms
= - 4x³ + 14x² - 36x + 45