Problem 73 - 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. $$ P(x)=8 x^{3}+10 x^{2}-39 x+9 ; \quad a=-3, b=2 $$

Step-by-Step Solution

Answer

The real zeros of the polynomial are bounded between \(-3\) and \(2\).

Step 1: Understand Polynomial and Bounds

We need to show that the polynomial \(P(x) = 8x^3 + 10x^2 - 39x + 9\) has its real zeros between \(a = -3\) and \(b = 2\). This means \(P(x)\) should change from positive to negative values or vice versa when evaluated at these bounds.

Step 2: Evaluate at Lower Bound

Calculate \(P(a) = P(-3)\). Substitute \(-3\) into the polynomial:\[P(-3) = 8(-3)^3 + 10(-3)^2 - 39(-3) + 9 = -216 + 90 + 117 + 9 = 0\]Since \(P(-3) = 0\), \(-3\) is a zero of the polynomial. However, not strictly greater or less than zero, it's a valid test point.

Step 3: Evaluate at Upper Bound

Now, calculate \(P(b) = P(2)\). Substitute \(2\) into the polynomial:\[P(2) = 8(2)^3 + 10(2)^2 - 39(2) + 9 = 64 + 40 - 78 + 9 = 35\]Since \(P(2) = 35 > 0\), the value is positive, suggesting \(2\) is not a zero but an upper bound.

Step 4: Establish Bounds

Since \(-3\) results in a value of zero and \(2\) results in a positive value, \(-3\) is an arbitrary lower bound with an actual zero, and \(2\) confirms an upper bound. It supports the claim that any real zeros of \(P(x)\) lie between these bounds.

Key Concepts

Real Zeros

Real zeros of a polynomial function are the x-values at which the function equals zero. In other words, these are the points where the graph of the polynomial intersects the x-axis. For a given polynomial like \( P(x) = 8x^3 + 10x^2 - 39x + 9 \), finding the real zeros involves determining the values of \( x \) that make \( P(x) = 0 \). In our exercise, we consider the polynomial values at specific points \( a = -3 \) and \( b = 2 \) to find such zeros. Real zeros are crucial because they offer insight into the behavior of the polynomial and give graphical points where the function changes signs.

Polynomial Evaluation

Evaluating a polynomial involves substituting a specific value of \( x \) into the polynomial expression and calculating the result. It's a simple procedure to test the behavior of a polynomial at specific x-values. In our exercise, we evaluate the polynomial \( P(x) = 8x^3 + 10x^2 - 39x + 9 \) at the points \( x = -3 \) and \( x = 2 \).

• For \( P(-3) \), substituting \( -3 \) gives the evaluation \( 0 \), confirming it as a zero of the polynomial.
• For \( P(2) \), substituting \( 2 \) yields \( 35 \), which is positive.

This evaluation helps us deduce that \( x = -3 \) is a zero, and \( x = 2 \) is not, indicating bounds for potential real zeros.

Upper and Lower Bounds

Finding upper and lower bounds for the real zeros of a polynomial is vital in narrowing down the number of potential zero candidates. In this exercise, the chosen values \( a = -3 \) and \( b = 2 \) help to establish these bounds effectively.
Lower bounds like \( a = -3 \) suggest that any real zero is either at this point or above it. Since \( P(-3) = 0 \), \( -3 \) is confirmed as an actual zero.
An upper bound \( b = 2 \) means any real zeros are below this value, evidenced by \( P(2) = 35 \), indicating the function is positive at this point. By choosing suitable bounds, we can more confidently limit our search for real zeros to this interval.

Polynomial Function Analysis

Analyzing a polynomial function consists of understanding its behavior by identifying zeros, evaluating certain values, and constraining real zero possibilities using bounds. This analysis assists in sketching the polynomial's curve to see where it changes direction or intersects with axes.
For the polynomial \( P(x) = 8x^3 + 10x^2 - 39x + 9 \), the analysis begins with evaluating real zeros and expanding to assess polynomial behavior near certain boundaries \( -3 \) and \( 2 \).

• Discovering \(-3\) as a zero helps us objectively analyze the polynomial’s root structure.
• Using \(x = 2\) as an evaluative point confirms that no additional zero lies beyond this, leading us into specialized strategies like synthetic division or graph plotting where needed.

Such analysis provides a broader picture of how the polynomial behaves over its domain, shedding light on critical value points for deeper exploration.