Problem 77 - Algebra and Trigonometry

Question

Find integers that are upper and lower bounds for the real zeros of the polynomial. $$ P(x)=x^{4}-2 x^{3}+x^{2}-9 x+2 $$

Step-by-Step Solution

Answer

Lower bound is -1 and upper bound is 3.

Step 1: Understand the problem

We need to find integers that serve as upper and lower bounds for the real zeros of the polynomial \( P(x) = x^4 - 2x^3 + x^2 - 9x + 2 \). This means determining the range in which the real roots of the polynomial exist.

Step 2: Identify the coefficients

List the coefficients of the polynomial \( P(x) = x^4 - 2x^3 + x^2 - 9x + 2 \). Here, the coefficients are 1, -2, 1, -9, and 2.

Step 3: Apply the Upper Bound Theorem

To find the upper bound, we apply the Upper Bound Theorem using synthetic division. We test positive integers to see when all the values in synthetic division are positive. We start with a larger integer and go downwards.

Step 4: Test integer for the upper bound

Using synthetic division, let's test the integer 2:- Perform synthetic division of \( P(x) \) by \( x-2 \).- If all the result's coefficients are non-negative, then 2 is an upper bound.Perform synthetic division:\[\begin{array}{r|rrrrr}2 & 1 & -2 & 1 & -9 & 2 \ & & 2 & 0 & 2 & -14 \\hline & 1 & 0 & 1 & -7 & -12 \\end{array}\]The coefficients are not all non-negative, so 2 is not an upper bound. Let's test 3 in the next step.

Step 5: Test another integer for upper bound

Test the integer 3 using synthetic division:- Perform synthetic division of \( P(x) \) by \( x-3 \).Perform synthetic division:\[\begin{array}{r|rrrrr}3 & 1 & -2 & 1 & -9 & 2 \ & & 3 & 3 & 12 & 9 \\hline & 1 & 1 & 4 & 3 & 11 \\end{array}\]All the coefficients are non-negative, so 3 is an upper bound.

Step 6: Apply the Lower Bound Theorem

To find the lower bound, we apply the Lower Bound Theorem using synthetic division. We test negative integers to find when values alternate in sign.

Step 7: Test integer for the lower bound

Using synthetic division, test the integer -1:- Perform synthetic division of \( P(x) \) by \( x+1 \).Perform synthetic division:\[\begin{array}{r|rrrrr}-1 & 1 & -2 & 1 & -9 & 2 \ & & -1 & 3 & -4 & 13 \\hline & 1 & -3 & 4 & -13 & 15 \\end{array}\]The sign of coefficients alternates, meaning -1 is a valid lower bound.

Step 8: Conclusion

The integer 3 serves as an upper bound and -1 serves as a lower bound for the real zeros of the polynomial.

Key Concepts

Polynomial Division

Polynomial division is a mathematical technique similar to the long division you might have learned in arithmetic, but it's applied to polynomials. It's especially useful for dividing a polynomial by a factor, helping determine if a certain value is a root or as part of finding bounds. In this approach, synthetic division is often used when dividing polynomials of the form \( (x - c) \), where \( c \) is a constant. Synthetic division is preferred over long division for its simplified framework, making it quicker and more efficient. It involves aligning the coefficients of the polynomial you're dividing by and performing operations that are similar to long division but with fewer steps and using only coefficients. For example, in our case, we applied synthetic division to check the upper and lower bounds by testing integers as potential zeros of the polynomial through either positive or negative results.

Upper Bound Theorem

The Upper Bound Theorem is utilized to find an integer that will be an upper boundary for the real zeros of a polynomial. If you suspect an integer \( k \) as an upper bound, you can test it by performing synthetic division of the polynomial with \( (x - k) \). If all the resulting coefficients from the division are non-negative (zero or positive), then the tested integer is an upper bound. This means that any zeros of the polynomial cannot be greater than \( k \). In our exercise, when testing \( x = 3 \), all coefficients were non-negative, confirming \( 3 \) as a valid upper bound.

Lower Bound Theorem

The Lower Bound Theorem, like the upper bound theorem, helps determine a boundary but for the smallest possible real zero of the polynomial. When you've identified a candidate integer \( c \) that could serve as the lower bound, perform synthetic division using \( (x + c) \). For \( c \) to be a lower bound, the coefficients must alternate in sign. This indicates that any zero of the polynomial is not smaller than \( c \). In our worked example, performing synthetic division on \( x = -1 \) resulted in alternating signs, thereby confirming it as a valid lower bound for the zeros.

Integer Bounds for Polynomials

Integer bounds are crucial in narrowing down the region where the real zeros of a polynomial are likely to be found. They effectively help students and mathematicians estimate potential real solutions without dealing with complex calculations. By using integer bounds, you apply the upper and lower bound theorems, simplifying finding zeros to a search within a specific interval. This interval, identified using synthetic division and the criteria for signs (either all positive or alternating), directs focus and reduces the manual searching for solutions. The combination of finding both the upper and lower bounds allows you to bracket the zeros, making solving the polynomial more manageable and systematic.