Question

List the potential rational zeros of the polynomial function \(P(x)=2 x^{3}-5 x^{2}+x-10\).

Short answer

The potential rational zeros are ±1, ±2, ±5, ±10, and ±1/2.

Step-by-step solution

- Identify the leading coefficient and the constant term

The leading coefficient is the coefficient of the highest degree term in the polynomial. Here, the leading coefficient is the coefficient of the term with the highest power of x, which is 3 in this case. The constant term is the term without any variables. In P(x) = 2x^3 - 5x^2 + x - 10, the leading coefficient is 2, and the constant term is -10.

- List the factors of the constant term

The constant term in the given polynomial is -10. List all the factors of -10: \ Factors of -10 are: \ \ \(\text{±1, ±2, ±5, ±10}\).

- List the factors of the leading coefficient

The leading coefficient in the given polynomial is 2. List all the factors of 2: \ Factors of 2 are: \ \ \(\text{±1, ±2}\).

- Form the list of potential rational zeros

Potential rational zeros of the polynomial are given by the formula \(\frac{\text{Factors of the constant term}}{\text{Factors of the leading coefficient}}\). Compute all possible combinations: \ \ \(\frac{1}{1}, \frac{2}{1}, \frac{5}{1}, \frac{10}{1}, \frac{-1}{1}, \frac{-2}{1}, \frac{-5}{1}, \frac{-10}{1}\) \(\frac{1}{2}, \frac{2}{2}, \frac{5}{2}, \frac{10}{2}, \frac{-1}{2}, \frac{-2}{2}, \frac{-5}{2}, \frac{-10}{2}\) \ Simplify the fractions to get: \ \ \(\text{±1, ±2, ±5, ±10, ±1/2}\).

Key concepts

leading coefficient

The leading coefficient is crucial when analyzing polynomials. It's the coefficient of the term with the highest power of the variable. For instance, in the polynomial function given in the problem, \(P(x) = 2x^3 - 5x^2 + x - 10\), the leading coefficient is 2, because it is the coefficient of the term with the highest exponent (which is \(2x^3\)).Identifying the leading coefficient helps in forming the potential rational zeros as part of the Rational Root Theorem, an essential tool in polynomial analysis.

constant term

In any polynomial, the constant term is the term that does not contain any variables. It's a standalone number. For the polynomial \(P(x) = 2x^3 - 5x^2 + x - 10\), the constant term is -10.The constant term is pivotal when applying the Rational Root Theorem. You will list its factors as part of determining the potential rational zeros of the polynomial function.

rational root theorem

The Rational Root Theorem is a fundamental tool in finding potential rational zeros of a polynomial. It states that for a polynomial \(P(x) = a_n x^n + ... + a_1 x + a_0\) with integer coefficients, any potential rational zero,\( \frac{p}{q}\), must be a fraction where \(p\) is a factor of the constant term (\(a_0\)), and \(q\) is a factor of the leading coefficient (\(a_n\)).Applying this theorem involves:

• Listing all factors of the constant term.
• Listing all factors of the leading coefficient.
• Forming fractions by dividing each factor of the constant term by each factor of the leading coefficient.
• Simplifying these fractions to get the potential rational zeros.

This method simplifies the process of solving polynomials by narrowing down the possible rational solutions we must test.

polynomial factors

Polynomials are expressions made up of variables and coefficients. A polynomial factorizes into components that, when multiplied together, give the original polynomial. For example, the polynomial \(P(x) = 2x^3 - 5x^2 + x - 10\) can be broken down into factors that, when solved, give potential rational zeros.To factorize a polynomial, especially using the Rational Root Theorem, you:

• Identify the leading coefficient and constant term.
• List their factors.
• Determine all possible fractions as potential rational zeros.
• Test these zeros in the polynomial to see if they solve the equation \(P(x) = 0\).

Understanding polynomial factors is key to solving polynomial functions efficiently and finding their roots.