Question - Using the Rational Zeros Theorem to Find Roots of a Polynomial
Solution:
The Rational Zeros Theorem states that if a polynomial has integer coefficients and has a rational zero $$ \frac{p}{q} $$, then $$ p $$ is a factor of the constant term and $$ q $$ is a factor of the leading coefficient.For the polynomial $$ 2c^3 - 10c^2 + 4c + 16 = 0 $$, the leading coefficient is 2, and the constant term is 16.The factors of 2 (the leading coefficient) are $$\pm 1, \pm 2$$, and the factors of 16 (the constant term) are $$\pm 1, \pm 2, \pm 4, \pm 8, \pm 16$$.Using the Rational Zeros Theorem, we can create a list of all the possible rational zeros for the polynomial by dividing the factors of the constant term by the factors of the leading coefficient. This gives us the potential rational zeros:$$ \pm 1, \pm 2, \pm 4, \pm 8, \pm 16, \pm \frac{1}{2}, \pm \frac{2}{2}, \pm \frac{4}{2}, \pm \frac{8}{2}, \pm \frac{16}{2} $$Reducing any fractions, the list simplifies to:$$ \pm 1, \pm 2, \pm 4, \pm 8, \pm 16, \pm \frac{1}{2}, \pm \frac{4}{2}, \pm \frac{8}{2} $$Which further simplifies to:$$ \pm 1, \pm 2, \pm 4, \pm 8, \pm 16, \pm \frac{1}{2}, \pm 2, \pm 4 $$Now that we have our possible rational zeros, we need to test them in the polynomial to see which, if any, are actual zeros. You can use synthetic division or direct substitution to test each rational zero.I cannot compute the solution to this polynomial equation, but you can follow these steps to systematically test each possible zero until you find a rational root. Once you find one root, you can factor it out and use the Factoring Theorem to factor the polynomial further, hence finding other roots. If you find a rational root, let's say $$ r $$, you will be able to divide the original polynomial by $$ (c - r) $$ to get a reduced polynomial of the second degree, which you can solve by factoring, completing the square, or using the quadratic formula.
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