Problem 51 - Algebra for College Students

Question

Establish the best integral bounds for the roots of each equation according to the theorem on bounds. $$x^{2}+x-13=0$$

Step-by-Step Solution

Answer

The roots are bounded between \(-4\) and \(4\).

Step 1: Identify the coefficients

The given quadratic equation is \(x^2 + x - 13 = 0\). Identify the coefficients: \(a = 1\), \(b = 1\), and \(c = -13\).

Step 2: Apply the Upper Bound Theorem

To establish the upper bound, we need to find a positive number such that all coefficients in the polynomial, when synthetic division is performed with that number, either remain non-negative or are strictly positive. The polynomial for this equation is \(1x^2 + 1x - 13\). Use trial and error to find the smallest positive integer that works.

Step 3: Check for upper bounds

Test \(x = 4\) as a potential upper bound: \(4^2 + 4 - 13 = 16 + 4 - 13 = 7\), which is positive. Therefore, an appropriate upper bound is \(x = 4\).

Step 4: Apply the Lower Bound Theorem

To establish the lower bound, find a negative number such that all coefficients in the polynomial, when synthetic division is performed with that number, change sign alternatively (i.e., positive to negative or negative to positive). Test with negative integers starting from \(-1\).

Step 5: Check for lower bounds

Test \(x = -4\): \((-4)^2 + (-4) - 13 = 16 - 4 - 13 = -1\), which follows the rule of changing the sign of coefficients. Therefore, an appropriate lower bound is \(x = -4\).

Step 6: Conclusion

According to the theorem on bounds, the roots of the equation \(x^2 + x - 13 = 0\) are bounded between \(-4\) and \(4\).

Key Concepts

Integral Bounds

Integral bounds in mathematics help us to understand where the roots of a polynomial lie. For the quadratic equation \(x^2 + x - 13 = 0\), integral bounds are established based on rules derived from the Upper and Lower Bound Theorems.

In simpler terms:

• Upper Bound: The smallest positive number that makes all results from synthetic division non-negative.


• Lower Bound: The largest negative number that makes results from synthetic division switch signs alternately.

Let’s see how these bounds are found and applied to this specific equation.

Upper Bound Theorem

The Upper Bound Theorem helps to find the highest range limit for the roots. For a given polynomial, it identifies the smallest positive integer that makes all results in synthetic division non-negative.

Here's how it was applied to \(x^2 + x - 13 = 0\):

• Identify coefficients: \(a = 1\), \(b = 1\), and \(c = -13\).


• Using trial and error, we test positive integers.


• For \(x = 4\), calculate \(4^2 + 4 - 13 = 16 + 4 - 13 = 7\).


• Since all results remain non-negative, the upper bound is 4.

This means that all roots of the polynomial are less than or equal to 4.

Lower Bound Theorem

The Lower Bound Theorem determines the highest negative range limit for the roots of a polynomial. It identifies the largest negative integer where synthetic division alternates signs in its results.

For the quadratic equation \(x^2 + x - 13 = 0\), we applied the theorem as follows:

• Test negative integers sequentially.


• For \(x = -4\), calculate \((-4)^2 + (-4) - 13 = 16 - 4 - 13 = -1\).


• The results change signs as needed: 16 (positive), -4 (negative), -13 (negative), resulting in -1 (negative).

Since the signs alternate correctly, the lower bound is determined to be -4. This means that all the roots are greater than or equal to -4.