Question

Find all zeros of the polynomial function or solve the given polynomial equation. Use the Rational Zero Theorem, Descartes's Rule of Signs, and possibly the graph of the polynomial function shown by a graphing utility as an aid in obtaining the first zero or the first root. 4x4−x3+5x2−2x−6=0

Accepted Answer

Identify the polynomial equation: $$4x^{4}$$ - $$x^{3}$$ + $$5x^{2} - 2x - 6 = 0$. Apply the Rational Zero Theorem to list all possible rational zeros. These are of the form $$\pm \frac{p}{q}$$, where p$ divides the constant term -6$ and q$ divides the leading coefficient 4$. So, possible values of p$ are $$\pm1, \pm2, \pm3, \pm6$$ and possible values of q$ are $$\pm1, \pm2, \pm4$$. Use Descartes's Rule of Signs to estimate the number of positive and negative real zeros. Count the sign changes in f(x$$)$$ for positive zeros and in f(-x$$)$$ for negative zeros to narrow down the possibilities. Test the possible rational zeros from step 2 by substituting them into the polynomial or using synthetic division to find a root that makes the polynomial equal to zero. Once a root is found, perform polynomial division (either synthetic or long division) to divide the original polynomial by the corresponding factor (x - r$$)$$, reducing the polynomial's degree. Then repeat the process on the resulting polynomial to find all zeros.

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Key Concepts

Rational Zero Theorem

Descartes's Rule of Signs

Graphing Polynomial Functions