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Mathematics, Tmm0022
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Real Zeros of Polynomials - Rational Zero Theorem, Descarte's Rule of Signs

Finding Real Zeros without Calculator

This section may be omitted. However, the methods and results in this topic are interesting enough for the curious student. It helps in promoting intuition about zeros of a polynomial. As well, this section helps reinforce the synthetic division method and solving quadratic equations. Further, it would be helpful to use technology to supplement this section.

In this module we will learn about finding real zeros of a polynomial using analytical techniques that primarily use synthetic division method. Cauchy's bounds give an interval in which all real zeros, if they exist, can be found. These bounds can be made sharper by using Upper and Lower Bounds Theorem. The Rational Zeros Theorem lists all possible rational numbers that could be potential candiates for zeros. Descarte's Rule of Signs helps in determining the different possibilities for positive and negative real zeros. The synthetic division method is used here to check which of these possibilities will be zeros.

Review: Synthetic Division Method and be able to recognize when a real number is a zero of a polynomial. Once a zero is found remaining zeros are found from the resulting quotient function. If the quotient function is quadratic then the rest of the zeros can be found.

Learning Objectives: Thoroughly understand the results of this Chapter and apply them

  • Cauchy's bounds for real zeros
  • Apply Rational Zero Theorem to find possible rational zero candidates. Use synthetic division to find identify such a zero and find the quotient. If the quotient is quadratic then the rest of the zeros can be found using appropriate methods of solving quadratic equations. If the quotient is not qaudratic, the process is find a zero is repeated.
  • Apply Descarte's Rule of Signs
  • Apply Upper and Lower Bound Theorem