Problem 71 - 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 lower and upper bounds, respectively, for the real zeros of \(P(x)\).

Step 1: Evaluate the Polynomial at \(a = -3\)

To determine if \(a = -3\) is a lower bound, we need to evaluate \(P(-3)\). Substituting \(-3\) into the polynomial, we have:\[P(-3) = 2(-3)^3 + 5(-3)^2 + (-3) - 2\]Calculating each term:- \(2(-3)^3 = 2(-27) = -54\)- \(5(-3)^2 = 5(9) = 45\)- \( (-3) = -3 \)- \( -2 = -2 \)Adding these results, we get:\[P(-3) = -54 + 45 - 3 - 2 = -14\]Since \(P(-3) < 0\), \(a = -3\) is a lower bound.

Step 2: Evaluate the Polynomial at \(b = 1\)

To verify if \(b = 1\) is an upper bound, evaluate \(P(1)\). Substituting \(1\) into the polynomial, we calculate:\[P(1) = 2(1)^3 + 5(1)^2 + (1) - 2\]Computing each term:- \(2(1)^3 = 2(1) = 2\)- \(5(1)^2 = 5(1) = 5\)- \( (1) = 1 \)- \( -2 = -2 \)Adding these values, we find:\[P(1) = 2 + 5 + 1 - 2 = 6\]As \(P(1) > 0\), \(b = 1\) is an upper bound.

Key Concepts

Real Zeros

Real zeros of a polynomial are the values of \(x\) that make the polynomial equal to zero. In simpler terms, they are the points on a graph where the curve intersects the x-axis. Finding these zeros is crucial because they help us understand the behavior and shape of the polynomial function.

One way to determine the potential existence of real zeros in an interval is by testing lower and upper bounds. If a polynomial switches signs, from positive to negative or vice versa, as you move across a certain interval, then it indicates the presence of a real zero. This is because the polynomial has to cross the x-axis to switch signs.

For example, if you plug in a lower bound \(a\) and the result \(P(a)\) is negative, and then plug in an upper bound \(b\) and \(P(b)\) is positive, there's a real zero between \(a\) and \(b\). This is based on the Intermediate Value Theorem, which ensures that continuous functions (like polynomials) take every value between \(P(a)\) and \(P(b)\).

Evaluating Polynomials

Evaluating a polynomial involves substituting a specific value for \(x\) in the polynomial expression and performing the arithmetic calculations. This is a straightforward, step-by-step process.

Consider evaluating the polynomial \(P(x) = 2x^3 + 5x^2 + x - 2\) at a given value. Begin by substituting the chosen \(x\) value into the polynomial. Then calculate each term separately:

• For the term \(2x^3\), multiply 2 by the cube of the given \(x\).
• For the term \(5x^2\), multiply 5 by the square of the given \(x\).
• Add the result of \(x\) itself.
• Subtract 2 as it is constant for any \(x\).

Once you have all these calculated values, add them together to get the final result of the polynomial evaluation at \(x\). This will give you a number indicating the value of the polynomial at that specific \(x\).

Ensuring accurate calculations during evaluation helps in correctly determining the behavior and bounds for potential real zeros.

Upper and Lower Bounds

Determining the upper and lower bounds of real zeros is an essential part of analyzing polynomials. These bounds give us a range within which all the real zeros of a polynomial lie. This approach significantly simplifies the process of discovering zeros.

To establish these bounds, we evaluate the polynomial at different values and observe the sign of the result. For instance, if evaluating \(P(x)\) at a point yields a negative number, and evaluating it at another point gives a positive number, it implies that there's a real zero between these two points. The number with the negative result acts as the lower bound, while the positive result indicates an upper bound.

• If \(P(a) < 0\), then \(a\) is a candidate for a lower bound, especially if higher values yield positive results.
• If \(P(b) > 0\), \(b\) may serve as an upper bound, especially if lower values yield negative results.

This method not only helps in finding zeros but also provides a clearer picture of the polynomial's behavior across a certain range. It ensures a systematic and organized approach to solving polynomial equations by focusing on specific intervals.