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A polynomial expression is a mathematical phrase that can include numbers, variables, and exponents. It is composed of terms separated by addition or subtraction signs. Each term can have:

• A coefficient, which is a numerical factor.
• A variable, such as \(x\).
• An exponent, which indicates the power to which the variable is raised.

For example, in the expression \(x^2 + x + 1\), \(x^2\) is a term where \(x\) is raised to the second power. The expression itself is a polynomial because it has multiple terms.
The polynomial's degree is determined by the highest power of the variable present. In our original exercise, the degree of the expression \(x^2 + x + 1\) is 2. Understanding these components helps you in operations such as multiplication and addition of polynomials.

The distributive property is a key rule in algebra that helps us multiply polynomials efficiently. It states that for any three terms \(a\), \(b\), and \(c\), the expression \(a(b+c)\) is equal to \(ab + ac\). This rule is crucial when dealing with polynomials.
For instance, let's apply this to our example: multiplying \((x-1)\) by \((x^2 + x + 1)\). According to the distributive property, each term in the first polynomial must be multiplied by each term in the second polynomial. This step-by-step multiplication ensures that all combinations are covered.
For our example, this becomes \(x(x^2) + x(x) + x(1) - 1(x^2) - 1(x) - 1(1)\). It’s important to pay attention to signs when applying the distributive property, particularly with subtraction involved.

After using the distributive property and expanding the expressions, you're often left with multiple terms. Some of these terms may be like terms, which are terms with the same variables raised to the same exponents.

• For instance, \(x^2\) and \(-x^2\) are like terms because they both contain \(x\) raised to the power of 2.
• Only their coefficients differ, making them combineable.

In our exercise, after simplifying the multiplication, we had: \(x^3 + x^2 + x - x^2 - x - 1\). Combining the like terms involves adding or subtracting their coefficients:

• Combine \(x^2 - x^2\), which becomes 0.
• Combine \(x - x\), which also cancels out to 0.

Thus, the simplified polynomial becomes \(x^3 - 1\). Understanding this process of identifying and combining like terms is essential for simplifying polynomial expressions effectively.