91Ó°ÊÓ

Polynomial equations are expressions that set a polynomial equal to zero. Polynomials are mathematical expressions involving variables and coefficients, using operations of addition, subtraction, multiplication, and non-negative integer exponents.
They are usually written in the form of \[a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0 = 0\]
where \(a_n, a_{n-1}, ..., a_1, a_0\) are constants, and \(n\) is a non-negative integer indicating the highest power of \(x\) in the expression.
In our example, the polynomial equation is \(1 - x + x^2 - x^3 = 0\). It is a cubic polynomial because the highest power of \(x\) is three. A root of this polynomial is any value for \(x\) that makes the equation true (equal to zero). Understanding this concept is crucial, as it allows us to identify solutions by testing various possible values for \(x\).

The substitution method is a simple technique used to determine if a specific value is a solution to an equation. In this method, you replace the variable in the equation with the value you are testing.
For example, if the equation is \(1 - x + x^2 - x^3 = 0\) and we want to check if \(x = -1\) is a root, we substitute \(-1\) for every occurrence of \(x\) in the equation. Each part of the equation involving \(x\) is evaluated separately.
In this exercise, substitution was done as follows:

• First term: remains constant as \(1\).
• Second term: substituting \(-1\) gives \(-(-1)\), simplifying to 1.
• Third term: \((-1)^2\) is 1.
• Fourth term: \(-(-1)^3\) simplifies to 1.

This method ensures that we correctly evaluate whether each term contributes to the overall solution, helping us see if the substituted value satisfies the equation.

Evaluating expressions involves performing operations to get a final numerical value. In the context of checking if a value is a root of an equation, evaluating the expression following substitution tests whether the substituted value satisfies the equality.
After substituting our test value into the equation \(1 - x + x^2 - x^3\), each term must be computed:

• First term: remains as \(1\).
• Second term: becomes \(1\) after simplification.
• Third term: evaluates to \(1\).
• Fourth term: also results in \(1\).

The sum of these evaluated terms was \(1 + 1 + 1 + 1 = 4\), which is not equal to zero. This indicates that \(x = -1\) is not a root of the equation. Evaluating expressions accurately is vital as it allows us to verify if the equation holds true under certain conditions, thus confirming or disproving potential solutions.