“I’m guessing that all of the real-number zeros are in the interval [𝒂, 𝒃]”
Use Synthetic Division to check your guesses
Testing your upper bound guess, b
Use Synthetic Division to divide 𝑃(𝑥) ÷ (𝑥 − 𝑏).
𝑏
𝑎𝑛
*
𝑞𝑛−1
⋯
*
⋯
𝑎1
*
𝑞0
𝑎0
*
𝑟
Coefficients of 𝑃(𝑥)
Synthetic division
Coefficients of the
Quotient and Remainder
 If the remainder 𝒓 = 𝟎, then
 𝑏 is a zero of 𝑃(𝑥)
 if we’re still interested in an upper bound, we must guess a higher value of 𝑏 and repeat
the testing procedure.
 But if the remainder 𝒓 ≠ 𝟎, check the signs of all the bottom row results, all of 𝑞𝑛−1 ⋯, 𝑞0 , 𝑟
 If they’re all ≥ 0, then YES, , 𝑏 is an upper bound. Any real-number zeros of 𝑃(𝑥) must be
≤ 𝑏.
 Otherwise, your guess is wrong. The upper bound is higher than what you guessed.
Testing your lower bound guess, a
Use Synthetic Division to divide 𝑃(𝑥) ÷ (𝑥 − 𝑎). (Note: the 𝑎 name for our upper bound is unrelated to
the 𝑎𝑠𝑢𝑏𝑠𝑐𝑟𝑖𝑝𝑡 names of the coefficients of 𝑃(𝑥). An unfortunate naming coincidence.)
𝑎
𝑎𝑛
*
𝑞𝑛−1
⋯
*
⋯
𝑎1
*
𝑞0
𝑎0
*
𝑟
Coefficients of 𝑃(𝑥)
Synthetic division
Coefficients of the
Quotient and Remainder
 If the remainder 𝒓 = 𝟎, then
 𝑎 is a zero of 𝑃(𝑥)
 if we’re still interested in a lower bound, guess a lower value of 𝑎 and retest.
 But if the remainder 𝒓 ≠ 𝟎, check the signs of all the bottom row results, all of 𝑞𝑛−1 ⋯, 𝑞0 , 𝑟
 If the signs on the bottom row, 𝑞𝑛−1 ⋯, 𝑞0 , 𝑟 alternate between positive and negative,
+, −, +, −, ⋯ or −, +, −, +, ⋯, then YES, 𝑎 is a lower bound. Any real-number zeros of
𝑃(𝑥) must be ≥ 𝑎.
 , then YES, , 𝑏 is an upper bound. Any real-number zeros of 𝑃(𝑥) must be ≤ 𝑏.
 Otherwise, your guess is wrong. The lower bound is lower than what you guessed.
Typical Guesses


𝑎 = −10 and 𝑏 = 10, because those are the WINDOW Xmin and Xmax values you get from the
TI-84 ZOOM 6:ZStandard command.
If you use the Rational Zero Theorem, there is a largest and smallest candidate for rational
number zeros, the ± of the constant term. Those can be good candidates for the bounded
zeros analysis.
Document1
2/6/2016 10:59 PM - D.R.S.