Question

Find all the real zeros of the function. \(g(x)=3 x^3-25 x^2+58 x-40\)

Short answer

The real roots of the polynomial \(g(x)=3 x^3-25 x^2+58 x-40\) are \(1\), \(2\), and \(5\).

Step-by-step solution

Apply the Rational Root Theorem

This theorem states that if a polynomial has a rational root, \(p/q\), then \(p\) divides the constant term (in this case 40) and \(q\) divides the leading coefficient (in this case 3). This means that the potential rational roots of this polynomial are \(±1, ±2, ±4, ±5, ±8, ±10, ±20, ±40, ±1/3, ±2/3\).

Test the Potential Rational Roots

Substitute each of the potential rational roots into the function and see if it evaluates to zero. Testing these values, we find that \(1\), \(2\), and \(5\) are roots of the function.

Re-write polynomial in factored form

Now that we know the roots of the polynomial we can write it in its factored form: \(g(x)=3(x-1)(x-2)(x-5)\).

Key concepts

Rational Root Theorem

The Rational Root Theorem is an invaluable tool when dealing with polynomial equations, especially when you are tasked with finding rational zeros. It's like a roadmap that gives you potential directions to explore. Here's how it works: if you're dealing with a polynomial of the form \(a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0\), it suggests that any rational solution \(\frac{p}{q}\) must fulfill two conditions.

• The numerator \(p\) must be a factor of the constant term \(a_0\).
• The denominator \(q\) must be a factor of the leading coefficient \(a_n\).

This helps us hone in on a limited set of possible rational roots, which we can then test. For example, in our polynomial \(g(x) = 3x^3 - 25x^2 + 58x - 40\), since the constant term is 40 and the leading coefficient is 3, our candidates are \(\pm 1, \pm 2, \pm 4, \pm 5, \pm 8, \pm 10, \pm 20, \pm 40, \pm \frac{1}{3}, \pm \frac{2}{3}\). By quickly testing these values, we can identify which, if any, are actual roots.

Polynomial Factoring

Factoring a polynomial is like breaking a complex machine into simple, digestible components. Once rational roots are discovered through the Rational Root Theorem, they can be used to factor the polynomial. This involves expressing the polynomial as a product of simpler polynomials.
For the polynomial \(g(x) = 3x^3 - 25x^2 + 58x - 40\), once we found the roots \(x = 1, 2, 5\), it becomes straightforward to factor. We express \(g(x)\) as \(3(x - 1)(x - 2)(x - 5)\). Factoring not only gives us a cleaner representation but also allows us to solve for roots effortlessly. Simply solve the factors set to zero, and you get the solution for \(x\). It also mirrors the algebraic structure and behavior of the function, which is particularly useful for graphing and understanding its properties.

Polynomial Functions

Polynomial functions are a cornerstone in algebra. They encompass expressions that combine constants, variables, and exponents into succinct mathematical models. The function \(g(x) = 3x^3 - 25x^2 + 58x - 40\) is a perfect example, illustrating how powerful these expressions can be in encapsulating complex relationships.

• Degree: The highest exponent (3, in this case) tells us the polynomial's degree, influencing its basic shape and the number of potential real roots.
• Coefficient: These numbers (like 3, -25, 58, -40 in the original polynomial) determine the function's growth rate and direction.

Such functions are versatile, found everywhere from physics to economics, describing phenomena that change non-linearly. Understanding their zeros—points where the function equals zero—gives critical insights into their behavior, like when an event occurs or when a quantity reaches equilibrium. Mastering them sets the stage for more advanced mathematical studies and applications.