SOLUTION: Complete the following: a) Find all zeros of f(x). b) Write the complete factored form of f(x). f(x) = x^4 + 5x^2 + 4

Question 58903This question is from textbook College Algebra with Modeling and Visualization
: Complete the following:
a) Find all zeros of f(x).
b) Write the complete factored form of f(x).
f(x) = x^4 + 5x^2 + 4 This question is from textbook College Algebra with Modeling and Visualization

Answer by uma(370) (Show Source): You can put this solution on YOUR website!
f(x) = x^4 + 5x^2 + 4
Let x^4 + 5x^2 + 4 = 0..............(1)
Let x^2 = y, then x^ 4 = y^2
So eqn (1) becomes y^2 + 5y + 4 = 0
==> y^2 + y + 4y + 4 = 0 [Splitting the middle term]
==> y(y+1) + 4(y+1) = 0
==> (y+1) (y+4) = 0
==> y+1 = 0 or y+4 = 0
==> y = -1 [adding -1 to both the sides] or y = -4 [Adding -4]
==> x^2 = -1 or x^2 = -4 [because y = x^2]
==> x = sqrt(-1) or x = sqrt(-4) [taking sqrt]
We know that sqrt(-1) = i
Therefore sqrt(-1) = i 0r -i and sqrt(-4) = 2i or -2i
Thus the zeroes of the given function are i, -i, 2i and -2i
f(x) = x^4 + 5x^2 + 4
= (x+i)(x-i)(x+2i)(x-2i)
Regards,
Uma

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