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1. A polynomial function is shown below.
f(x) = x^6 + 12x^5 + 43x^4 + 22x^3
a. Find the real zeros of f (x).
b. Express f (x) in terms of linear factors.
c. Find all of the zeros of f (x).
(3 points)
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First, factor by taking x^3 out:
f(x) = = .
Next, check that x = -1 is the root of the polynomial in parentheses .
So, the polynomial in parentheses has the factor (x+1).
Make long division and factor this polynomial:
= .
The quadratic polynomial x^2 + 11x + 32 has no real roots (since its discriminant = -7 is negative).
Thus you can not factor this quadratic polynomial further over real domain.
Therefore, your final factoring over the real domain is
f(x) = = .
The two real roots are x = 0 and x = -1.
The complex zeroes of the polynomial x^2 + 11x + 32 are
= and = .