Find a polynomial of degree 3 with real coefficients that satisfies the given conditions.
-Zeros of $- 5 , i , - i$ and $P ( - 3 ) = 60$
A) $P ( x ) = 3 x ^ { 3 } + 15 x ^ { 2 } + 3 x + 15$
B) $P ( x ) = - 3 x ^ { 3 } - 15 x ^ { 2 } - 3 x - 15$
C) $P ( x ) = - 3 x ^ { 3 } - 15 x ^ { 2 } + 3 x + 15$
D) $P ( x ) = 3 x ^ { 3 } + 15 x ^ { 2 } - 3 x - 15$

Question

Quizplus · Accepted Answer

Since the zeros are $ - 5 , i , - i$, the polynomial can be written as:$$P(x) = a(x + 5)(x - i)(x + i)$$$$= a(x^2 + 4x + 5)(x^2 + 1)$$$$= a(x^4 + 5x^3 + 9x^2 + 4x + 5)$$To find the value of $a$, we use the given condition $P(-3) = 60$:$$P(-3) = a((-3)^4 + 5(-3)^3 + 9(-3)^2 + 4(-3) + 5) = 60$$$$a(81 - 135 + 81 - 12 + 5) = 60$$$$a(140) = 60$$$$a = \frac{60}{140} = \frac{3}{7}$$Therefore, the polynomial is:$$P(x) = \frac{3}{7}(x^4 + 5x^3 + 9x^2 + 4x + 5)$$$$= \frac{3}{7}x^4 + \frac{15}{7}x^3 + \frac{27}{7}x^2 + \frac{12}{7}x + \frac{15}{7}$$$$= 3x^4 + 15x^3 + 3x^2 + 15$$

Find a Polynomial of Degree 3 with Real Coefficients That −5,i,−i- 5 , i , - i−5,i,−i

Find a Polynomial of Degree 3 with Real Coefficients That $- 5 , i , - i$