Use the Binomial Theorem to expand and then simplify the result: (x² +x+ 1)³.

Verified step by step guidance

1. Recognize that the Binomial Theorem applies to expressions of the form \((a + b)^n\). Since the given expression is \((x^2 + x + 1)^3\), which has three terms, we cannot directly apply the Binomial Theorem. Instead, we will expand it by multiplying the expression by itself three times: \((x^2 + x + 1)(x^2 + x + 1)(x^2 + x + 1)\).
2. First, multiply the first two polynomials: \((x^2 + x + 1)(x^2 + x + 1)\). Use the distributive property (FOIL method) to multiply each term in the first polynomial by each term in the second polynomial.
3. After multiplying, combine like terms to simplify the result of the first multiplication. This will give you a quadratic expression in terms of \(x\).
4. Next, multiply the simplified result from step 3 by the third polynomial \((x^2 + x + 1)\), again using the distributive property to multiply each term.
5. Finally, combine like terms from the multiplication in step 4 to get the fully expanded and simplified form of \((x^2 + x + 1)^3\).

Key Concepts

Binomial Theorem

The Binomial Theorem provides a formula to expand expressions raised to a power, specifically for binomials of the form (a + b)^n. It uses binomial coefficients, often found in Pascal's Triangle, to determine the coefficients of each term in the expansion.

Polynomial Expansion

Polynomial expansion involves multiplying expressions to remove parentheses and combine like terms. When expanding powers of polynomials with more than two terms, such as trinomials, repeated multiplication and careful combination of like terms are necessary.

Combining Like Terms

After expanding an expression, like terms—terms with the same variable raised to the same power—must be combined by adding their coefficients. This simplification step is essential to write the polynomial in its simplest form.