Problem 65 - Algebra and Trigonometry

Question

Show that the given values for \(a\) and \(b\) are lower and upper bounds for the real zeros of the polynomial. $$ P(x)=2 x^{3}+5 x^{2}+x-2 ; \quad a=-3, b=1 $$

Step-by-Step Solution

Answer

The values \( a = -3 \) and \( b = 1 \) are the respective lower and upper bounds for the real zeros of the polynomial.

Step 1: Identify the Polynomial

We have the polynomial function \( P(x) = 2x^3 + 5x^2 + x - 2 \). Our goal is to verify if \( a = -3 \) is a lower bound and \( b = 1 \) is an upper bound for the real zeros of this polynomial.

Step 2: Apply the Lower Bound Theorem

Use the Lower Bound Theorem, which states if \( -a_0 \) (a negative number) is a lower bound of a polynomial, then in synthetic division with \(-a_0\), all the coefficients of the quotient and the remainder must have alternating signs. We will perform synthetic division of \(P(x)\) by \(x + 3\) to check this condition.

Step 2.1: Perform Synthetic Division for Lower Bound Check

Place the coefficients \( [2, 5, 1, -2] \) for polynomial division by \(x + 3\) (which means using -3). Perform synthetic division:1. Bring down the leading coefficient: 2.2. Multiply -3 by 2, add to next coefficient: \( 5 + (-6) = -1 \).3. Multiply -3 by -1, add to next coefficient: \(1 + 3 = 4 \).4. Multiply -3 by 4, add to next coefficient: \(-2 - 12 = -14 \).Quotient is [2, -1, 4] with remainder -14.

Step 2.2: Check Signs in Result of Synthetic Division

The resulting sequence of 2, -1, 4, -14 shows that the signs alternate perfectly (positive -> negative -> positive -> negative). Thus, \( a = -3 \) is confirmed as a lower bound.

Step 3: Apply the Upper Bound Theorem

Use the Upper Bound Theorem, which states if a positive number \( b_0 \) is an upper bound of a polynomial, then in synthetic division with \( b_0 \), all the coefficients of the quotient and the remainder must be non-negative. We will perform synthetic division of \(P(x)\) by \(x - 1\) to check this condition.

Step 3.1: Perform Synthetic Division for Upper Bound Check

Place the coefficients \( [2, 5, 1, -2] \) for polynomial division by \(x - 1\) (which means using 1). Perform synthetic division:1. Bring down the leading coefficient: 2.2. Multiply 1 by 2, add to next coefficient: \( 5 + 2 = 7 \).3. Multiply 1 by 7, add to next coefficient: \(1 + 7 = 8 \).4. Multiply 1 by 8, add to next coefficient: \(-2 + 8 = 6 \).Quotient is [2, 7, 8] with remainder 6.

Step 3.2: Verify Non-Negative Coefficients

All values in the quotient and remainder are non-negative: 2, 7, 8, 6. Thus, \( b = 1 \) is confirmed as an upper bound.

Key Concepts

Lower Bound Theorem

The Lower Bound Theorem gives us a way to determine whether a given number is a lower bound for the real zeros of a polynomial. To use this theorem, we employ synthetic division.
When checking if a negative number, like \(-3\), is a lower bound, you perform synthetic division of the polynomial by \(x + 3\). The goal is to have the coefficients of the quotient and the remainder alternate in sign. This means each number in the result should switch between positive and negative as you go along.
In our specific example:

• The polynomial coefficients are \([2, 5, 1, -2]\).
• We divide by \(x + 3\) using synthetic division.
• The results are \([2, -1, 4, -14]\), which alternate perfectly.

Thus, \(a = -3\) is successfully confirmed as a lower bound because the sequence of numbers alternates between positive and negative. This means no real zero of the polynomial is less than \(-3\).

Upper Bound Theorem

To identify an upper bound for the real zeros of a polynomial, we use the Upper Bound Theorem. This theorem works with synthetic division as well, but with a positive number instead.
If \(b_0\) is an upper bound, then all the values from the synthetic division should be non-negative. In short, everything should be zero or a positive number.
For our function, we are testing \(b = 1\) by performing synthetic division using \([2, 5, 1, -2]\) divided by \(x - 1\). The process goes as follows:

• First coefficient is simply brought down: 2.
• Next, multiply 1 by 2 and add to the next coefficient: \(5 + 2 = 7\).
• Multiply 1 by 7 and add: \(1 + 7 = 8\).
• Finally, multiply 1 by 8 and add to the last coefficient: \(-2 + 8 = 6\).

The quotient is \([2, 7, 8]\) with remainder 6, all non-negative. Thus, \(b = 1\) qualifies as an upper bound, meaning no real zero of the polynomial is greater than \(1\).

Synthetic Division

Synthetic division is a method used to divide polynomials by polynomials of the form \(x - c\), where \(c\) is a constant. It's a faster and simpler alternative to traditional long division in algebra. The basics involve using the coefficients of the polynomial, aligning them, and performing a series of multiplications and additions.
To effectively utilize synthetic division:

• Write down the coefficients of the polynomial excluding any other terms.
• Equate the divisor to zero to find \(c\); if dividing by \(x + 3\), switch it to \(-3\).
• Start with the left-most coefficient and bring it straight down.
• Multiply \(c\) by the number brought down, adding this product to the next coefficient.
• Repeat until completing the polynomial.

Using our example: when checking \(a = -3\) as a potential lower bound, with \(c = -3\), perform synthetic division with coefficients \([2, 5, 1, -2]\). This calculation helps determine alternating signs for a lower bound. Similarly, for an upper bound with \(b = 1\), the results are all non-negative, confirming its correctness.
Synthetic division not only verifies bounds but is also a powerful technique for simplifying polynomial remainder calculations and determining polynomial factor relationships.