Problem 88 - Precalculus Mathematics for Calculus

Question

Show that the polynomial does not have any rational zeros. $$P(x)=x^{50}-5 x^{25}+x^{2}-1$$

Step-by-Step Solution

Answer

The polynomial has no rational zeros.

Step 1: Understanding the Problem

We need to determine if the polynomial \( P(x) = x^{50} - 5x^{25} + x^2 - 1 \) has any rational zeros. According to the Rational Root Theorem, any rational root, \( \frac{p}{q} \), where \( p \) is a factor of the constant term and \( q \) is a factor of the leading coefficient, must satisfy the equation.

Step 2: Identifying Factors

The constant term of \( P(x) \) is \(-1 \) and the leading coefficient is \(1\). The factors of \(-1\) are \( \pm 1\), and the factors of \(1\) are just \( \pm 1\). Therefore, according to the Rational Root Theorem, the only possible rational roots are \( 1 \) and \(-1 \).

Step 3: Checking Potential Rational Zeros

Substitute each possible rational root into the polynomial to see if it equals zero. First, try \( x = 1 \):\[ P(1) = 1^{50} - 5 \cdot 1^{25} + 1^2 - 1 = 1 - 5 + 1 - 1 = -4. \] Now, try \( x = -1 \):\[ P(-1) = (-1)^{50} - 5 \cdot (-1)^{25} + (-1)^2 - 1 = 1 + 5 + 1 - 1 = 6. \] Neither \( 1 \) nor \(-1 \) is a root of the polynomial.

Step 4: Conclusion

Since neither \( 1 \) nor \(-1 \) is a root of the polynomial, and these are the only potential rational zeros as per the Rational Root Theorem, the polynomial \( P(x) \) has no rational zeros.

Key Concepts

Polynomial Roots

Polynomial roots are the values of the variable that make the polynomial equal to zero. For any polynomial, its roots can be rational, irrational, or complex. Understanding the roots is vital for solving polynomials and can help in factoring them. When determining the roots, we can use different theorems and techniques based on the specific polynomial given. In this exercise, we focused on identifying rational roots using the Rational Root Theorem. Rational roots are the simplest to check because they are presentable in the form of a fraction. Let's get the concept straight:

• Roots: Solutions where the polynomial equals zero.
• Types: Rational, Irrational, Complex.
• Understanding: Simplifies solving and factoring polynomials.

Finding the roots of a high-degree polynomial analytically can be quite challenging but knowing which potential rational forms can exist saves time and effort. It's crucial to remember that not all polynomials have rational roots, and as in our exercise example, sometimes none satisfy the polynomial equation, confirming their absence.

Factorization

Factorization involves breaking down a polynomial into simpler polynomials whose product is the original polynomial. It relies on identifying roots or zeros and expressing the polynomial as a product of factors tied to these roots. Understanding factorization can make solving polynomials and algebraic expressions easier. Here, the presence or absence of rational roots informs factorization.

• Simplification: Turns complex expressions into products of simpler terms.
• Efficiency: Aids in evaluating large polynomials quickly.
• Shortcuts: Knowledge of criteria like the Rational Root Theorem reduces effort in determining factors.

In cases like our exercise, knowing that the polynomial has no rational roots indicates that it cannot be factored over the rationals. Hence, one would have to explore irrational or complex numbers if further factorization was needed.

Rational Numbers

Rational numbers are numbers that can be expressed as the quotient or fraction of two integers, where the denominator is not zero. They hold a significant place in mathematics, especially when examining polynomial roots.Understanding rational numbers is vital when using the Rational Root Theorem. This theorem applies to identify potential rational roots among the coefficients of the polynomial.

• Components: Can be written as \( \frac{p}{q} \), where \( p \) and \( q \) are integers, and \( q eq 0 \).
• Significance: Helps in testing potential roots quickly.
• Application: Forms the basis for the Rational Root Theorem.

In our exercise, by identifying the rational numbers derived from the factors of the constant and leading coefficient, we quickly narrowed our list of potential roots to test for the polynomial equation.