Problem 87 - Algebra and Trigonometry

Question

Show that the polynomial does not have any rational zeros. $$ P(x)=3 x^{3}-x^{2}-6 x+12 $$

Step-by-Step Solution

Answer

The polynomial has no rational zeros.

Step 1: Identify Potential Rational Zeros Using the Rational Root Theorem

The Rational Root Theorem states that any potential rational root, expressed as \( \frac{p}{q} \), must have \( p \) as a factor of the constant term (12) and \( q \) as a factor of the leading coefficient (3). Therefore, the possible values for \( p \) are ±1, ±2, ±3, ±4, ±6, ±12, and for \( q \) they are ±1, ±3. Thus, the possible rational roots are ±1, ±2, ±3, ±4, ±6, ±12, ±\( \frac{1}{3} \), ±\( \frac{2}{3} \), ±\( \frac{4}{3} \), ±\( \frac{6}{3} \), ±\( \frac{12}{3} \).

Step 2: Test Each Potential Rational Zero

Substitute each potential rational zero determined in Step 1 into the polynomial \( P(x) = 3x^3 - x^2 - 6x + 12 \) to check if it equals zero. For instance, substitute \( x = 1 \): \( P(1) = 3(1)^3 - (1)^2 - 6(1) + 12 = 8 \), which is not zero. Repeat this process for each potential rational root.

Step 3: Show That None of the Values Satisfy the Polynomial Equation

Upon checking each value, you'll find that none of the potential rational roots make \( P(x) = 0 \). Values such as ±1, ±2, ±3, ±4, ±6, ±12, ±\( \frac{1}{3} \), ±\( \frac{2}{3} \), ±\( \frac{4}{3} \), ±\( \frac{6}{3} \), and ±\( \frac{12}{3} \) all yield non-zero results when substituted into \( P(x) \).

Step 4: Conclude That the Polynomial Has No Rational Zeros

Since none of the potential rational roots actually satisfy the equation \( P(x) = 0 \), we can conclude that the polynomial \( 3x^3 - x^2 - 6x + 12 \) does not have any rational zeros.

Key Concepts

Polynomial Functions

Polynomial functions are mathematical expressions consisting of variables, coefficients, and a combination of addition, subtraction, multiplication, and non-negative integer exponents of variables. They can be simple, like a linear polynomial, or complex, like higher-degree polynomials. In our exercise example, the polynomial is given as:\[P(x) = 3x^3 - x^2 - 6x + 12\]This is a third-degree polynomial function because the highest power of the variable \(x\) is 3. Polynomial functions are fundamental in algebra and calculus, and they play a significant role in understanding mathematical relationships and modeling real-world phenomena.

• The degree of a polynomial is determined by the highest exponent on its variable.
• A polynomial of degree \(n\) can have up to \(n\) roots.
• Polynomials can be expressed graphically as smooth, continuous curves.

Rational Zeros

Rational zeros, also known as rational roots, are the solutions to a polynomial equation that can be expressed as a fraction \(\frac{p}{q}\), where \(p\) and \(q\) are integers and \(q eq 0\). The Rational Root Theorem helps determine potential rational zeros of a polynomial function.
In the exercise, the Rational Root Theorem was used to list all potential rational zeros of the polynomial \(P(x) = 3x^3 - x^2 - 6x + 12\). It states that for a polynomial with integer coefficients, any rational solution \(\frac{p}{q}\) must have \(p\) as a factor of the constant term and \(q\) as a factor of the leading coefficient.

• Potential rational zeros are often tested individually to determine if they actually satisfy the polynomial equation \(P(x)=0\).
• The Rational Root Theorem simplifies the process of finding rational zeros by reducing the number of potential candidates.

Factorization

Factorization is the process of breaking down a polynomial into simpler, irreducible components called factors. These factors multiply together to give the original polynomial. Finding factors can help determine the zeros of a polynomial.
In this exercise, however, it was found that no rational zeros exist for the polynomial, implying it cannot be conveniently factored into linear factors with rational coefficients. Because every root of a factor corresponds to a zero in the polynomial, the absence of rational zeros suggests that factorization requires complex or irrational numbers.

• Polynomials can often be factored into linear, quadratic, or other polynomial expressions.
• When a polynomial has no rational zeros, factorization may involve higher mathematical methods beyond basic algebra.

Leading Coefficient

The leading coefficient of a polynomial is the coefficient of the term with the highest degree. In our given polynomial \(P(x) = 3x^3 - x^2 - 6x + 12\), the leading coefficient is 3.
The leading coefficient influences the graph's end behavior. A positive leading coefficient with an odd degree results in the graph rising to the right and falling to the left, while a negative leading coefficient results in the opposite. Furthermore, during the application of the Rational Root Theorem, the leading coefficient is used to determine potential rational zeros.

• The leading coefficient can affect the steepness of the graph and the polynomial's behavior as \(x\) approaches infinity or negative infinity.
• Altering the leading coefficient changes the polynomial but maintains its degree.