Problem 77 - Precalculus Mathematics for Calculus

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

The lower bound is any negative integer less than -2, and the upper bound is -2.

Step 1: Identify the degree of the polynomial

The polynomial given is \(P(x) = x^4 - 2x^3 + x^2 - 9x + 2\). It is a 4th degree polynomial because the highest power of \(x\) is 4.

Step 2: Determine the bounds using the Rational Root Theorem

For the polynomial \(P(x) = x^4 - 2x^3 + x^2 - 9x + 2\), the potential rational zeros are factors of the constant term, 2. These factors are \(\pm 1, \pm 2\).

Step 3: Evaluate the polynomial at potential root bounds

Calculate \(P(x)\) for each possible rational root to check each as an upper or lower bound. We evaluate \(P(1)\), \(P(2)\), \(P(-1)\), \(P(-2)\).

Step 4: Calculate \(P(1)\)

Evaluate the function: \(P(1) = 1^4 - 2(1)^3 + 1^2 - 9(1) + 2 = 1 - 2 + 1 - 9 + 2 = -7\). Since \(P(1) < 0\), 1 is not an upper bound.

Step 5: Calculate \(P(2)\)

Evaluate the function: \(P(2) = 2^4 - 2(2)^3 + 2^2 - 9(2) + 2 = 16 - 16 + 4 - 18 + 2 = -12\). Since \(P(2) < 0\), 2 is not an upper bound.

Step 6: Calculate \(P(-1)\)

Evaluate the function: \(P(-1) = (-1)^4 - 2(-1)^3 + (-1)^2 - 9(-1) + 2 = 1 + 2 + 1 + 9 + 2 = 15\). Since \(P(-1) > 0\), \(-1\) is an upper bound.

Step 7: Calculate \(P(-2)\)

Evaluate the function: \(P(-2) = (-2)^4 - 2(-2)^3 + (-2)^2 - 9(-2) + 2 = 16 + 16 + 4 + 18 + 2 = 56\). Since \(P(-2) > 0\), \(-2\) is an upper bound.

Step 8: Confirm the bounds using Descartes' Rule of Signs

According to Descartes' Rule of Signs, there is one sign change in \(P(x)\), which means there is one positive real root. And there are three sign changes in \(P(-x) = x^4 + 2x^3 + x^2 + 9x + 2\), so there are either 3 or 1 negative real roots. Therefore, \(-2\) is confirmed as a suitable upper bound.

Key Concepts

Real Zeros

Real zeros refer to the points where the graph of a polynomial function crosses or touches the x-axis on a coordinate plane. Essentially, these are the solutions to the equation when the polynomial is set equal to zero. Finding these zeros helps in understanding the behavior of the polynomial and how it interacts with its graph.
When looking for real zeros, the goal is to solve the equation of the polynomial \[ P(x) = 0. \] Real zeros can be found by factoring the polynomial or using other algebraic methods such as the Rational Root Theorem or Descartes' Rule of Signs, which will help to identify potential candidates for real zeros. In the given polynomial, \[ x^4 - 2x^3 + x^2 - 9x + 2 = 0, \] estimating the bounds of real zeros involves checking possible rational solutions and confirming their validity through systematic evaluation.

Rational Root Theorem

The Rational Root Theorem is a valuable tool for finding potential zeros of a polynomial that are rational numbers. It states that if a polynomial has integer coefficients, any rational solution, expressed as a fraction \(\frac{p}{q}\), must have:

• the numerator \(p\) as a factor of the constant term
• the denominator \(q\) as a factor of the leading coefficient.

In simple terms, you list all the possible fractions made from these factors, and these fractions are potential rational roots.
For the polynomial \[ P(x) = x^4 - 2x^3 + x^2 - 9x + 2, \] the constant term is 2, and the leading coefficient (of \(x^4\)) is 1. Hence, the potential rational roots are \(\pm1\) and \(\pm2\). These numbers are then tested in the polynomial to determine if they are real zeros.

Descartes' Rule of Signs

Descartes' Rule of Signs provides a way to predict the number of positive and negative real roots in a polynomial. According to the rule:

• The number of positive real roots of a polynomial is equal to the number of sign changes between consecutive non-zero coefficients, or is less than it by a multiple of two.
• To find the number of negative real roots, replace each \(x\) with \(-x\) in the polynomial and count the sign changes.

Applying this to the polynomial \[ P(x) = x^4 - 2x^3 + x^2 - 9x + 2, \] we see one sign change, indicating one positive real root. Substituting \(-x\) for \(x\) gives \[ P(-x) = x^4 + 2x^3 + x^2 + 9x + 2, \] which has three sign changes, suggesting there are either 3 or 1 negative real roots. This analysis helps define potential bounds and validate actual real zero findings.