Question

Find all the zeros of the function and write the polynomial as a product of linear factors. $$f(t)=t^{3}-3 t^{2}-15 t+125$$

Short answer

The only zero of the polynomial \(t^{3}-3 t^{2}-15 t+125\) is \(t = 5\), and the polynomial can be written as a product of linear factors as \(f(t) = (t-5)(t-5)(t-5)\) or simplified as \(f(t) = (t-5)^{3}\).

Step-by-step solution

Identify the polynomial

The polynomial given in the problem is \(f(t)=t^{3}-3 t^{2}-15 t+125\). We're asked to find its zeros, i.e., the values of \(t\) for which \(f(t)\) equals to zero.

Factorize the Polynomial

Factorizing the polynomial will involve identifying a common factor, or using the factor theorem or rational root theorem to identify potential factors. After some experimentation and attempts, find that \(f(t) = t^{3}-3 t^{2}-15 t+125 = (t-5)^{3}\).

Write the polynomial as a product of linear factors

To write the polynomial as a product of the linear factors, consider that each factor gives a root of the polynomial, so the general form of the polynomial in terms of linear factors is \(f(t) = (t - x_{1})(t - x_{2})(t - x_{3})\), where \(x_{1}, x_{2}, x_{3}\) are the roots of the polynomial. In our case, the polynomial becomes \(f(t) = (t-5)(t-5)(t-5)\), since the only root of the polynomial is \(t = 5\)

Find the Zeros of the Polynomial

The zeros of the polynomial are the solutions to the equation \(f(t) = 0\), i.e., the values of \(t\) for which the polynomial equals zero. After factoring, we can see that the equation \((t-5)(t-5)(t-5) = 0\) is only true when \(t = 5\). Therefore, 5 is the only zero of the polynomial.

Key concepts

Factor Theorem

Understanding the Factor Theorem is like having a secret key that unlocks the door to finding the roots of a polynomial. It states that if a value, say \(a\), makes the polynomial \(f(x)\) equal to zero (i.e., \(f(a) = 0\)), then \(x-a\) is a factor of that polynomial. In other words, every root of the polynomial corresponds to a linear factor.

For our polynomial, \(f(t) = t^{3} - 3t^{2} - 15t + 125\), we applied this theorem. First, we checked various integers to see if they satisfy the condition \(f(a) = 0\) and found that \(f(5) = 0\). This immediately tells us that \(t-5\) is a factor of \(f(t)\). However, in this case, not only is \(t-5\) a factor, but the entire polynomial can be written as \(f(t) = (t-5)^{3}\), meaning the factor \(t-5\) is repeated three times, which is somewhat unusual but possible in polynomials.

Linear Factors of Polynomial

When we talk about linear factors of a polynomial, we mean the simplest forms of factors that can't be broken down any further, namely those of the form \(x-c\), where \(c\) represents a root of the polynomial. Each linear factor corresponds to a zero of the polynomial, and when a polynomial is broken down completely, it can be expressed as a product of these linear factors.

In the context of the given polynomial, \(f(t) = (t-5)(t-5)(t-5)\), we can see the polynomial is expressed in terms of a single linear factor raised to the third power, indicating that it only has one unique zero, which is 5. Generally, a polynomial of degree \(n\) will have \(n\) linear factors (including repetitions), providing a full breakdown of the original polynomial into the product of its roots.

Rational Root Theorem

If you've been mystified trying to figure out possible rational zeros of a polynomial, the Rational Root Theorem can serve as a guiding light. This theorem suggests that if a polynomial has rational roots, they are among the fractions formed by the divisors of the constant term and the leading coefficient. For a polynomial \(f(x) = a_{n}x^n + \cdots + a_{0}\), any potential rational zero, \(\frac{p}{q}\), must satisfy that \(p\) is a factor of the constant term \(a_{0}\), and \(q\) is a factor of the leading coefficient \(a_{n}\).

In our exercise, the constant term is 125 and the leading coefficient is 1, which makes it straightforward. The divisors of 125 are ±1, ±5, ±25, ±125. Since our leading coefficient is 1, we don't have to consider any other numbers for \(q\) in \(\frac{p}{q}\), simplifying the search for rational roots considerably. This tool is incredibly helpful when the polynomial doesn't factor easily or has multiple terms, and one must propose possible rational zeros to test using the Factor Theorem.