Problem 14 - Precalculus: Functions and Graphs

Question

Show that the equation has no rational root. $$2 x^{5}+3 x^{3}+7=0$$

Step-by-Step Solution

Answer

The equation \(2x^5 + 3x^3 + 7 = 0\) has no rational root because none satisfy the equation when tested.

Step 1: Understanding Rational Root Theorem

The Rational Root Theorem states that any rational root of a polynomial equation with integer coefficients must be of the form \( p/q \), where \( p \) is a factor of the constant term and \( q \) is a factor of the leading coefficient.

Step 2: Identify factors of the constant term and leading coefficient

The constant term of the polynomial \( 2x^5 + 3x^3 + 7 = 0 \) is 7, which has factors \( \pm 1, \pm 7 \). The leading coefficient is 2, which has factors \( \pm 1, \pm 2 \).

Step 3: List all possible rational roots

Based on Step 2, the possible rational roots are \( \pm 1, \pm 7, \pm \frac{1}{2}, \pm \frac{7}{2} \).

Step 4: Test each possible rational root

Substitute each possible rational root into the polynomial:1. For \( x = 1 \), \( 2(1)^5 + 3(1)^3 + 7 = 2 + 3 + 7 = 12 eq 0 \).2. For \( x = -1 \), \( 2(-1)^5 + 3(-1)^3 + 7 = -2 - 3 + 7 = 2 eq 0 \).3. For \( x = 7 \), \( 2(7)^5 + 3(7)^3 + 7 \) is much larger than zero.4. For \( x = -7 \), \( 2(-7)^5 + 3(-7)^3 + 7 \) is very negative.5. For \( x = \frac{1}{2} \), \( 2(\frac{1}{2})^5 + 3(\frac{1}{2})^3 + 7 eq 0 \).6. For \( x = -\frac{1}{2} \), \( 2(-\frac{1}{2})^5 + 3(-\frac{1}{2})^3 + 7 eq 0 \).7. For \( x = \frac{7}{2} \), \( 2(\frac{7}{2})^5 + 3(\frac{7}{2})^3 + 7 eq 0 \).8. For \( x = -\frac{7}{2} \), \( 2(-\frac{7}{2})^5 + 3(-\frac{7}{2})^3 + 7 eq 0 \).

Step 5: Conclusion

Since none of the possible rational roots make the polynomial equal to zero, the equation \( 2x^5 + 3x^3 + 7 = 0 \) has no rational roots.

Key Concepts

Polynomial equations

Polynomial equations are expressions that equate a polynomial to zero. A polynomial itself is a mathematical expression comprising variables, coefficients, and constants, arranged in terms raised to whole number powers. For example, in the polynomial equation \(2x^5 + 3x^3 + 7 = 0\), the highest power of the variable \(x\) is 5, indicating it's a fifth-degree polynomial.
There are several key parts to every polynomial:

• Terms: Each product of a coefficient and a variable raised to an exponent like \(2x^5\).
• Degree: The largest exponent, which determines the degree of the polynomial.
• Coefficients: The numbers multiplying the terms, such as 2 in \(2x^5\).
• Constant Term: A term that does not have a variable, like 7 in this case.

Solving such polynomial equations involves finding roots, i.e., values of \(x\) that make the expression equal zero.

Rational roots

Rational roots refer to solutions of a polynomial equation that can be expressed as a fraction. An essential tool for identifying rational roots is the Rational Root Theorem.
The Rational Root Theorem asserts that if a polynomial has a rational root \(\frac{p}{q}\), then:

• \(p\) (the numerator) is a factor of the constant term of the polynomial.
• \(q\) (the denominator) is a factor of the leading coefficient of the polynomial.

This theorem significantly narrows down the list of potential rational roots to check. It streamlines the process of determining whether any rational solution exists by providing a finite list to test.

Leading coefficient

The leading coefficient is the coefficient of the term with the highest degree in a polynomial. It plays a crucial role in the Rational Root Theorem.
In the equation \(2x^5 + 3x^3 + 7 = 0\), the leading coefficient is 2 because it's associated with \(x^5\), the term with the highest power. When using the Rational Root Theorem, the possible values for the denominator \(q\) in the rational root \(\frac{p}{q}\) must be factors of the leading coefficient. In this case, the factors of 2 are \(\pm 1\) and \(\pm 2\).
The leading coefficient also affects the behavior of the graph of the polynomial, influencing its end behavior and how the polynomial's roots impact the x-axis crossings.

Constant term

The constant term is the term in a polynomial that does not contain any variables; it remains constant as the values of variables change. In \(2x^5 + 3x^3 + 7 = 0\), the constant term is 7.
When determining possible rational roots using the Rational Root Theorem, the numerator \(p\) of any potential rational root \(\frac{p}{q}\) must be a factor of this constant term. For the polynomial in question, the factors of 7 are \(\pm 1\) and \(\pm 7\).
This term plays a key role in simplifying and limiting the potential numbers we need to check when identifying possible roots, thereby making the problem more manageable.