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Chapter 3: Problem 95

Find all zeros of \(f(x)=x^{3}+5 x^{2}-8 x+2\)

Short Answer

Expert verified
The roots of the function \(f(x) = x^3 + 5x^2 - 8x + 2\) are -1 (with multiplicity 4) and -2

Step by step solution

01

Apply Rational Root Theorem

The Rational Root Theorem states that any rational root, say \( p/q \) of a Polynomial equation \[ a_{n} x^n + a_{n鈭�1} x^{n鈭�1} + a_{n鈭�2} x^{n鈭�2} + . . . + a_2 x^2 + a_1 x + a_0 = 0 \] where \( a_{n}, a_{n鈭�1}, a_{n鈭�2}, . . . , a_2, a_1, a_0 \) are integers, should be such that p is a factor of \( a_0 \) and q is a factor of \( a_n \). Here \( a_0 = 2, a_n = 1 \). The factors of 2 are 卤1, 卤2 and factors of 1 are 卤1. Thus the possible rational roots of the given function are 卤1, 卤2.
02

Try possible rational roots

To find the real roots we can substitute the possible rational roots into the given polynomial. If the value of the polynomial becomes zero for any substitution, that value is a real root. By trial and error, \(x = -1\) and \(x= -2\) are both zeros of the polynomial.
03

Perform Polynomial Division

Next, we perform polynomial division by removing the factors corresponding to the roots -1 and -2. We start by dividing the polynomial by \( (x + 1) \) which leaves us with \( x^2 + 4x - 2 \). Then we divide \( x^2 + 4x - 2 \) by \( (x + 2) \) to get \( x^2 + 2x + 1 \) which factors to \( (x + 1)^2 \). Thus our polynomial can be rewritten as \( (x + 1)^3 \). The root of \( (x + 1)^2 \) is \( x = -1 \), but as it is a double root, this means that \( -1 \) is a root four times.

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