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The substitution method is a straightforward approach used to determine if a specific value is a zero of a polynomial function. To do this, you substitute the value into the polynomial and evaluate the resulting expression.
If the result is zero, the value is indeed a zero of the polynomial.

Let's revisit the polynomial given in the exercise:
\( f(x) = 2x^3 - 3x^2 + x + 6 \).
To check if a number is a zero of this polynomial, substitute the number into all instances of \( x \) in the equation and simplify:

• If \( f(2) = 0 \), then 2 is a zero of \( f(x) \).
• If \( f(3) = 0 \), then 3 is a zero of \( f(x) \).
• If \( f(-1) = 0 \), then -1 is a zero of \( f(x) \).

This method is reliable and can be applied to any polynomial function to verify its roots easily.

Polynomial functions are expressions involving variables raised to whole-number exponents and coefficients. The general form of a polynomial function is:
\( f(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0 \)
Here, \( a_n, a_{n-1}, \ldots, a_1, a_0 \) are constants, and \( x \) is the variable.

In the given exercise, the polynomial function is:

\( f(x) = 2x^3 - 3x^2 + x + 6 \)

Each term in this function represents a different degree of \( x \). The term \( 2x^3 \) has a degree of 3, \( -3x^2 \) has a degree of 2, \( x \) has a degree of 1, and \( 6 \) is the constant term (degree 0).

When solving polynomial functions, we often seek their zeros or roots, which are the values of \( x \) that make the function equal to zero. Finding the zeros helps understand the behavior of the polynomial graph and its intercepts with the x-axis.

Successfully solving math problems, especially those involving polynomial functions, requires a systematic approach. Here are strategies to follow:

• Understand the problem: Ensure you grasp what the problem is asking. Identify the function and the values to be tested.
• Plan the solution: Decide on the method, such as using substitution to test possible zeros in this case.
• Execute the plan: Substituting the values into the polynomial and simplifying step-by-step helps you find the results.
• Verify your answer: Confirm your computational steps. If \( f(x) \) equals zero for a certain value, your solution is correct.

Using these strategies improves problem-solving skills and builds confidence when working with polynomial functions and other math problems.

Additionally, documenting each step, as shown in the provided solution, helps track progress and ensures no details are overlooked.