91Ó°ÊÓ

A polynomial function is a mathematical expression that involves a sum of powers of variables. In simpler terms, it's a combination of terms like constants, variables, and exponents. The general form of a polynomial is given by:

\( f(x) = a_n x^n + a_{n-1} x^{n-1} + \ldots + a_1 x + a_0 \)

Here,

• \(a_n, a_{n-1}, \ldots, a_0 \) are constants called coefficients,



• \(x \) is the variable, and



• \(n \) (which must be a non-negative integer) determines the degree of the polynomial.

Advertisement Let's consider the polynomial \( f(x) = 4x^3 - 5x^2 - 3x + 1 \), from the exercise. Here, we can see:

• The highest power of \(x\) is 3, so it's a cubic polynomial.



• The leading coefficient (coefficient of the term with the highest power) is 4.



• The constant term (term without \(x\)) is 1.

Advertisement Understanding polynomial functions is crucial because they appear in various fields – from physics to economics.

Evaluation of polynomials simply means finding the value of the polynomial for a given value of \(x\). The process is straightforward – substitute the given value of \(x\) into the polynomial and perform the arithmetic operations.

In our exercise, we needed to evaluate \( f(x) = 4x^3 - 5x^2 - 3x + 1 \) at \( x = \frac{1}{3} \). Here’s how we did it:
1. Substitute \( x = \frac{1}{3} \) into the polynomial:
\( f\left( \frac{1}{3} \right) = 4 \left( \frac{1}{3} \right)^3 - 5 \left( \frac{1}{3} \right)^2 - 3 \left( \frac{1}{3} \right) + 1 \)
2. Simplify each term:
\( \left( \frac{1}{3} \right)^3 = \frac{1}{27} \)
\( \left( \frac{1}{3} \right)^2 = \frac{1}{9} \)
3. Apply these simplified values back into the polynomial:
\( f\left( \frac{1}{3} \right) = 4 \cdot \frac{1}{27} - 5 \cdot \frac{1}{9} - 3 \cdot \frac{1}{3} + 1 \)
\( = \frac{4}{27} - \frac{5}{9} - 1 + 1 \)
4. Simplify further:
\( = \frac{4}{27} - \frac{15}{27} - \frac{9}{27} + \frac{27}{27} \)
\( = \frac{-20 + 27}{27} = \frac{7}{27} \)
After evaluation, we see that the result is not zero.

The Rational Zeros Theorem is a useful tool for determining the possible rational zeros (also known as roots) of a polynomial. It states that any rational zero of a polynomial function \(f(x)\) must be a fraction \(p/q\), where:

• \(p\) is a factor of the constant term \(a_0\),



• \(q\) is a factor of the leading coefficient \(a_n\).

For the polynomial \(4x^3 - 5x^2 - 3x + 1\):

• The constant term \(a_0\) is 1, whose factors are \(\pm1\).



• The leading coefficient \(a_n\) is 4, whose factors are \(\pm1, \pm2, \pm4\).

Thus, the possible rational zeros are:

• \(p/q = \pm1, \pm\frac{1}{2}, \pm\frac{1}{4}\)

The theorem mandates that whenever we try a value from this list and it satisfies \(f(x) = 0 \), it's a zero of the polynomial. In this exercise, we tested \(x = \frac{1}{3}\). Although it's not one of the possible rational zeros as indicated by the Rational Zeros Theorem, try evaluating it we just showed us that \( x = \frac{1}{3}\) does not lead to zero.

In mathematics, a zero of a function is a value of \(x\) that makes the function equal to zero. In simpler terms, it's a solution to the equation \(f(x) = 0\). For polynomial functions, zeros are the values of \(x\) where the polynomial crosses or touches the x-axis.

To find the zeros of a polynomial function:

• Use the Rational Zeros Theorem to list possible rational zeros.



• Evaluate the polynomial at these possible zeros.



• If \( f(x) = 0\), that value is a zero of the polynomial.

In the exercise given, we checked if \(x = \frac{1}{3}\) was a zero of \(f(x) = 4x^3 - 5x^2 - 3x + 1\).

After evaluating the polynomial at \(x = \frac{1}{3}\) and getting \( \frac{7}{27}\) which is not zero, we concluded that \( \frac{1}{3}\) is not a zero. Finding zeros is essential because they tell us where the function hits the x-axis, giving us insights into polynomial behavior and roots.