Positive Real Zeros Of F(x) = 8x^4 - 72x^3 + 144x^2

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Determining the number of positive real zeros of a polynomial function is a fundamental concept in algebra. In this article, we will explore how to find the number of possible positive real zeros for the polynomial function f(x) = 8x^4 - 72x^3 + 144x^2. This involves understanding Descartes' Rule of Signs and applying it effectively. By the end of this guide, you’ll have a clear understanding of how to approach such problems and the underlying principles that make it work.

Understanding Descartes' Rule of Signs

Descartes' Rule of Signs is a powerful theorem that provides information about the nature of the roots of a polynomial. Specifically, it helps us determine the possible number of positive and negative real roots. The rule states:

  1. The number of positive real roots of a polynomial f(x) is either equal to the number of sign changes in f(x) or is less than that by an even number.
  2. The number of negative real roots of a polynomial f(x) is either equal to the number of sign changes in f(-x) or is less than that by an even number.

To apply this rule, we first need to understand what a sign change is. A sign change occurs when the coefficient of a term changes its sign (from positive to negative or vice versa) as we read the polynomial from left to right. Let's delve deeper into how this rule works and why it is so useful.

Sign Changes in f(x)

To identify sign changes, we focus on the coefficients of the polynomial terms. Consider the polynomial f(x) = ax^n + bx^(n-1) + cx^(n-2) + ... + k, where a, b, c, ..., k are coefficients. We look for changes in the sign of these coefficients. For example, if the coefficients go from positive to negative or from negative to positive, that counts as a sign change.

For the given polynomial f(x) = 8x^4 - 72x^3 + 144x^2, the coefficients are 8, -72, and 144. Let’s trace the signs:

  • The coefficient 8 is positive (+).
  • The coefficient -72 is negative (-).
  • The coefficient 144 is positive (+).

We observe that there are two sign changes: from +8 to -72 and from -72 to +144. According to Descartes' Rule of Signs, this means there are either two positive real roots or zero positive real roots (two less than two).

Sign Changes in f(-x)

To find the possible number of negative real roots, we need to analyze f(-x). This involves substituting x with -x in the polynomial and simplifying. For our polynomial, this is how it works:

f(-x) = 8(-x)^4 - 72(-x)^3 + 144(-x)^2

Now, let's simplify:

  • 8(-x)^4 = 8x^4 (since a negative number raised to an even power is positive)
  • -72(-x)^3 = +72x^3 (since a negative number raised to an odd power is negative, and a negative times a negative is positive)
  • 144(-x)^2 = 144x^2 (since a negative number raised to an even power is positive)

So, f(-x) = 8x^4 + 72x^3 + 144x^2. Now we analyze the signs of the coefficients, which are 8, 72, and 144. All coefficients are positive, so there are no sign changes. This indicates that there are no negative real roots for the polynomial.

Applying Descartes' Rule to f(x) = 8x^4 - 72x^3 + 144x^2

Now, let's systematically apply Descartes' Rule of Signs to determine the possible number of positive real zeros for the given polynomial function f(x) = 8x^4 - 72x^3 + 144x^2. This involves analyzing the sign changes in f(x) and f(-x).

Analyzing Sign Changes in f(x)

As we determined earlier, the polynomial f(x) = 8x^4 - 72x^3 + 144x^2 has coefficients 8, -72, and 144. The signs change from positive to negative (8 to -72) and then from negative to positive (-72 to 144). This gives us two sign changes.

According to Descartes' Rule of Signs, the number of positive real zeros is either equal to the number of sign changes or less than that by an even number. In this case, there are two sign changes, so the possible number of positive real zeros is either 2 or 2 - 2 = 0. Therefore, f(x) can have either two or zero positive real zeros.

Analyzing Sign Changes in f(-x)

We found that f(-x) = 8x^4 + 72x^3 + 144x^2. The coefficients are 8, 72, and 144, all of which are positive. There are no sign changes in f(-x), which means there are no negative real zeros for f(x).

Finding the Zeros Directly

To verify our findings using Descartes' Rule of Signs, we can attempt to find the zeros of the polynomial directly. This involves setting f(x) = 0 and solving for x.

f(x) = 8x^4 - 72x^3 + 144x^2 = 0

First, we can factor out the common term 8x^2:

8x2(x2 - 9x + 18) = 0

Now, we have two factors: 8x^2 and (x^2 - 9x + 18). Setting each factor to zero:

  1. 8x^2 = 0 implies x = 0 (a repeated root)
  2. x^2 - 9x + 18 = 0 can be factored further: (x - 3)(x - 6) = 0 This gives us two solutions: x = 3 and x = 6.

So, the zeros of f(x) are x = 0 (with multiplicity 2), x = 3, and x = 6. We have two distinct positive real zeros (3 and 6) and one real zero at 0. This confirms our findings from Descartes' Rule of Signs that there can be two positive real zeros.

Conclusion

In conclusion, by applying Descartes' Rule of Signs and directly finding the zeros of the polynomial f(x) = 8x^4 - 72x^3 + 144x^2, we determined that there are two possible positive real zeros. The rule provided us with a quick way to ascertain the nature of the roots without explicitly solving the polynomial equation. This approach is valuable in understanding the behavior of polynomial functions and their roots.

Understanding Descartes' Rule of Signs is an important tool in algebra for determining the possible number of positive and negative real roots of a polynomial. By analyzing the sign changes in f(x) and f(-x), we can gain insights into the nature of the roots without having to solve the polynomial equation explicitly. In the case of f(x) = 8x^4 - 72x^3 + 144x^2, we found that there are either two or zero positive real roots, and direct calculation confirmed the presence of two distinct positive real roots. This method is a crucial aspect of polynomial analysis and is applicable to various algebraic problems.

For further exploration on polynomial functions and their properties, you might find resources on trusted websites like Khan Academy Algebra helpful. This can provide additional insights and practice problems to enhance your understanding.