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Chapter 2: Problem 28

Use Descartes' Rule of Signs to state the number of possible positive and negative real zeros of each polynomial function. $$P(x)=x^{3}-19 x-30$$

Short Answer

Expert verified
The given function \(P(x) = x^3 - 19x - 30\) has 2 or 0 positive real zeros and 1 or 0 negative real zeros.

Step by step solution

01

Find the number of sign changes in the given polynomial

To find the number of positive zeros, write the polynomial in standard form and count the number of sign changes. The given polynomial is \( P(x) = x^{3} - 19x -30 \). It has two sign changes, from positive to negative for \(x^3\) to \(-19x\), and from negative to positive for \(-19x\) and \(-30\). Therefore, it has 2 or less positive real zeros.
02

Find the number of sign changes when x is replaced by -x

To find the number of negative zeros, replace \(x\) in the polynomial with \(-x\) and count the number of sign changes. Replacing \(x\) with \(-x\) in \(P(x)\) gives \( P(-x) = -x^{3} + 19x -30 \). This expression has one sign change, from negative to positive \(-x^3\) to \(19x\), therefore it has either 1 or less negative real zeros.

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