Problem 74 - Precalculus Mathematics for Calculus

Question

Show that the given values for \(a\) and \(b\) are lower and upper bounds for the real zeros of the polynomial. $$P(x)=3 x^{4}-17 x^{3}+24 x^{2}-9 x+1 ; \quad a=0, b=6$$

Step-by-Step Solution

Answer

0 and 6 are the bounds for real roots of the polynomial.

Step 1: Understand Bounds

We are given a polynomial \(P(x) = 3x^4 - 17x^3 + 24x^2 - 9x + 1\) and the task is to show that \(a = 0\) is a lower bound and \(b = 6\) is an upper bound for its real zeros. This involves showing that the polynomial does not have real roots less than \(a\) or greater than \(b\).

Step 2: Verify Lower Bound with Descartes' Rule of Signs

Descartes' Rule of Signs helps determine the number of positive and negative real zeros of a polynomial. Evaluate \(P(-x)\) to check the sign changes, indicating the number of real roots in \((-\infty, 0)\). Here, substitute \(-x\) to get: \[ P(-x) = 3(-x)^4 - 17(-x)^3 + 24(-x)^2 - 9(-x) + 1 = 3x^4 + 17x^3 + 24x^2 + 9x + 1 \]The number of sign changes comparing the coefficients is 0, indicating no real roots in \((-\infty, 0)\). Thus, \(a = 0\) is a lower bound.

Step 3: Verify Upper Bound with Descartes' Rule of Signs

To verify the upper bound \(b = 6\), we use synthetic division to evaluate \(P(x)\) for values greater than 0, like testing \(x = 6\). Perform synthetic division on \(P(x)\) using \(x = 6\):Coefficients of \(P(x)\) are \([3, -17, 24, -9, 1]\).Set up like this:```6 | 3 -17 24 -9 1 | 18 6 180 1029 ----------------------- 3 1 30 171 1030```The remainder is positive (1030) which implies no sign change and confirms \(b = 6\) as an upper bound for real roots.

Key Concepts

Descartes' Rule of Signs

Descartes' Rule of Signs is a powerful technique used to estimate the number of positive and negative real zeros of a polynomial. This rule provides an initial understanding of where the real zeros might lie within a certain range.
According to Descartes' Rule, the number of positive real zeros in a polynomial function corresponds directly to the number of times the sign of its coefficients changes, or is less by an even number. For finding negative real zeros, the same rule applies after substituting \(-x\) into the polynomial.

• To check for positive real zeros, look at the original polynomial's coefficients. Count the sign changes.
• For negative real zeros, substitute \(-x\) and count the sign changes in the new expression.

In the given exercise, this rule is used to confirm whether 0 can be a lower bound, by substituting \(-x\) for the polynomial, which revealed no change in signs, confirming no real roots exist below 0.

Synthetic Division

Synthetic division is a shortcut method for dividing a polynomial by a linear term of the form \(x - c\). It's particularly useful for checking if a given value is a root of the polynomial or if it's an upper or lower bound for the roots.
This process simplifies everything to working just with the coefficients of the polynomial, making it faster than traditional polynomial division.

• To perform synthetic division, arrange the polynomial coefficients and bring down the leading coefficient.
• Multiply it by the test value (in this case, 6) and add it to the next coefficient, continuing this pattern across all coefficients.
• The final result includes both a quotient and a remainder.

If all the coefficients in the quotient formed (excluding the remainder) are non-negative when testing for boundaries, it confirms that the chosen upper or lower bound does not contain real roots. In the textbook solution, using synthetic division with x=6 confirmed that 6 was an upper bound for real roots, as the remainder was positive, and there were no negative coefficients in the quotient.

Real Zeros

Real zeros, or real roots, of a polynomial refer to the values of \(x\) that satisfy the equation \(P(x) = 0\). These are the points where the graph of the polynomial intersects the x-axis.
Determining the real zeros is a foundational task in algebra, as it helps in understanding the behavior of polynomial functions. Real zeros might be bounded, meaning they lie within a certain range, which is often essential in many real-world applications.
In the context of the exercise, we are tasked with ensuring that real zeros do not exist below \(a = 0\) or above \(b = 6\). Using methods like Descartes' Rule and synthetic division helps establish these boundaries. Establishing bounds can help narrow down the possibilities and provide constraints when searching for any potential real zeros, thus simplifying more complex analyses or factorization of the polynomial.