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Unformatted text preview: ﬁnding the zeros and checking the sign of f using test values,
we are using test values to determine where the signs switch to ﬁnd the zeros. It is a slow and
tedious, yet fool-proof, method for determining an approximation of a real zero. Our last example
reminds us that ﬁnding the zeros of polynomials is a critical step in solving polynomial equations
1. Find all of the real solutions to the equation 2x5 + 6x3 + 3 = 3x4 + 8x2 .
2. Solve the inequality 2x5 + 6x3 + 3 ≤ 3x4 + 8x2 .
3. Interpret your answer to part 2 graphically, and verify using a graphing calculator. Solution.
1. Finding the real solutions to 2x5 + 6x3 + 3 = 3x4 + 8x2 is the same as ﬁnding the real
solutions to 2x5 − 3x4 + 6x3 − 8x2 + 3 = 0. In other words, we are looking for the real zeros
of p(x) = 2x5 − 3x4 + 6x3 − 8x2 + 3. Using the techniques developed in this section, we divide
We don’t use this word lightly; it can be proven that the zeros of some polynomials cannot be expressed using...
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