Download 9.3 Lower and Upper Bounds for Real Roots of Polynomial Equations

9.3
Real Zeros of Polynomial Functions
Lower and Upper Bounds for Real Roots of Polynomial Equations
At times, the list of possible rational roots for a polynomial equation is rather lengthy.
However, you can use patterns to shorten the list. One pattern involves finding lower and upper
bounds for real roots. If a number is an upper bound, then there are no real roots for the
equation greater than that number. If a number is a lower bound, then there are no real roots for
the equation lower than that number.
Consider the polynomial equation x3 + 3x2 − 34x − 42 = 0
1. List all the possible rational zeros for the equation.
When testing the possible positive rational roots, if the last row of numbers in the synthetic
division table are all positive, then the number is an upper bound.
2. Use synthetic division to see whether 6 is a root of the given polynomial.
Notice the last row contains all positive numbers. Therefore, 6 is an upper bound. We can
rule out any other numbers in our possible rational zeros list that are greater than 6.
3. Which numbers can be eliminated from our list of possible rational roots?
When testing the possible negative rational roots, if the last row of numbers in the synthetic
division table are of alternating signs, then the number is a lower bound.
4. Use synthetic division to see whether –14 is a root of the given polynomial.
Notice the last row contains numbers with alternating signs. Therefore, − 14 is a
lower bound. We can rule out any other numbers in our possible rational roots list
that are less than − 14.
5. Which numbers can be eliminated from our list of possible rational roots?
6. What numbers in our original list from Exercise 1 are still remaining?
7. Find all roots of the given polynomial.
8. List all possible rational roots for the polynomial equation
x 4  x3  30 x 2  36 x  216  0 . Then use synthetic division to find all zeros.
Note any upper or lower bounds.
9. List all possible rational roots for the polynomial equation
x 4  3x3  8 x 2  36 x  48  0 . Then use synthetic division to find all zeros. Note
any upper or lower bounds.
10. List all possible rational roots for the polynomial equation
x 4  4 x3  29 x 2  64 x  720  0 . Then use synthetic division to find all zeros.
Note any upper or lower bounds.