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# Descartes Rule of Signs Help (page 2)

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By McGraw-Hill Professional
Updated on Oct 4, 2011

## The Upper and Lower Bounds Theorem

The Upper and Lower Bounds Theorem helps us to find a range of x -values that will contain all real zeros. It does not tell us what these bounds are. We make a guess as to what these bounds are then check them. For a negative number x = a, the statement “ a is a lower bound for the real zeros” means that there is no number to the left of x = a on the x -axis that is a zero. For a positive number x = b, the statement “ b is an upper bound for the real zeros” means that there is no number to the right of x = b on the x-axis that is a zero. In other words, all of the x -intercepts are between a and b.

To determine whether a negative number x = a is a lower bound for a polynomial, we need to use synthetic division. If the numbers in the bottom row alternate between nonpositive and nonnegative numbers, then x = a is a lower bound for the negative zeros. A “nonpositive” number is 0 or negative, and a “nonnegative” number is 0 or positive.

To determine whether a positive number x = b is an upper bound for the positive zeros, again we need to use synthetic division. If the numbers on the bottom row are all nonnegative, then x = b is an upper bound on the positive zeros.

#### Examples

Show that the given values for a and b are lower, and upper bounds, respectively, for the following polynomials.

• f(x) = x 4 + x 3 − 16 x 2 − 4 x + 48; a = −5 and b = 5
• The bottom row alternates between positive and negative numbers, so a = −5 is a lower bound for the negative zeros of f(x) .

The entries on the bottom row are all positive, so b = 5 is an upper bound for the positive zeros of f(x) . All of the real zeros for f(x) are between x = − 5 and x = 5.

If 0 appears on the bottom row when testing for an upper bound, we can consider 0 to be positive. If 0 appears in the bottom row when testing for a lower bound, we can consider 0 to be negative if the previous entry is positive and positive if the previous entry is negative. In other words, consider 0 to be the opposite sign as the previous entry.

• P(x) = 4 x 4 + 20 x 3 + 7 x 2 + 3 x − 6 with a = −5

Because 0 follows a positive number, we will consider 0 to be negative. This makes the bottom row alternate between positive and negative entries, so a = −5 is a lower bound for the negative zeros of P(x).

### Limitations of the Upper and Lower Bounds Theorem

The Upper and Lower Bounds Theorem has some limitations. For instance, it does not tell us how to find upper and lower bounds for the zeros of a polynomial. For any polynomial, there are infinitely many upper and lower bounds. For instance, if x = 5 is an upper bound, then any number larger than 5 is also an upper bound. For many polynomials, a starting place is the quotient of the constant term and the leading coefficient and its negative: . First show that these are bounds for the zeros, then work your way inward. For example, if f(x) = 2 x 3 − 7 x 2 + x + 50, let and . Then, let a and b get closer together, say a = − 10 and b = 10.

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