Problem 81 - College Algebra

Question

Use the Binomial Theorem to expand and then simplify the result: \(\left(x^{2}+x+1\right)^{3}\) Hint: Write \(x^{2}+x+1\) as \(x^{2}+(x+1)\).

Step-by-Step Solution

Answer

The expansion and simplification of the expression \(\left(x^{2}+x+1\right)^{3}\) is \(x^{6} + 3*x^{5} + 9*x^{4} + 11*x^{3} + 15*x^{2} + 18*x + 6\)

Step 1: Understand the binomial theorem

The Binomial Theorem states that for 2 variables (a and b) and any given positive integer n, the equation (a+b)^n can be expanded into a series. Here, we need to apply the theorem to the given expression \(x^{2}+x+1\). We are going to arrange the expression as: \(x^{2}+(x+1)\) to match the binomial syntax (two terms), which are our a and b.

Step 2: Apply the Binomial Theorem to the given expression

Apply the binomial theorem with a= \(x^{2}\), b= \(x+1\) and n=3. Applying the theorem, we get the expanded form as below: \(\left( x^{2} \right) ^{3} + 3 \times \left( x^{2} \right) ^{2} \times (x + 1) + 3 \times x^{2} \times (x + 1) ^{2} + (x + 1) ^{3}\)

Step 3: Simplifying the expanded result

Next, each of the terms of the series is simplified as: \(x^{6} + 3 \times x^{4} \times (x + 1) + 3 \times x^{2} \times (x^{2}+2x+1) + (x^{3}+3x^{2}+3x+1)\) which can be further simplified as based on algebraic mathematics rules into: \(x^{6} + 3*x^{5} + 3*x^{4} + 6*x^{4} + 6*x^{3} + 3*x^{2} + x^{3} + 3*x^{2} + 3*x + 1\).

Step 4: Combine similar terms and get the final result

Finally, add all the similar terms to get the final expression. In the simplified binomial expression, \(x^{6}\) and \(x^{5}\) only have one of each. Add up \(x^{4}\), \(x^{3}\) , \(x^{2}\) and \(x\). This gives you the final expression: \(x^{6} + 3*x^{5} + 9*x^{4} + 11*x^{3} + 15*x^{2} + 18*x + 6\)

Key Concepts

Polynomial Expansion

Polynomial expansion is a process of expressing a polynomial raised to a power as a complete sum of its terms. In our exercise, we use the Binomial Theorem to expand \( \left( x^{2}+x+1 \right)^3 \). The theorem allows us to systematically calculate each term in the expanded expression.
To begin, identify the polynomial expression you wish to expand. In this case, that expression is \( x^{2}+(x+1) \). By treating \( x^{2} \) as one variable and \( x+1 \) as another, we can apply the Binomial Theorem effectively.

• First, breakdown the setup using the binomial expression formula, which requires breaking it into parts.
• Then, execute each term according to their binomial coefficients which are based on combinatorial principles.
• Finally, simplify each term as you expand it and prepare for combination.

Through polynomial expansion within a binomial, you unlock a series of terms you can then simplify into a neat, expanded, and complete expression.

Combinatorial Mathematics

Combinatorial mathematics plays a crucial role in the application of the Binomial Theorem due to its usage of binomial coefficients. These coefficients are determined using combinations, a principle from combinatorial mathematics, which provides the number of ways to select items from a set.
When expanding \( \left( a+b \right)^n \) using the Binomial Theorem, each term is attributed a specific coefficient, computed with combinations: \(\binom{n}{k}\). Here, \(n\) is the exponent, and \(k\) changes from 0 to \(n\).

• These coefficients tell us how many times each arrangement of the terms occurs.
• Using them allows for effective and accurate expansion into the correct polynomial form.
• In our exercise, the values of \(n=3\) and the sum of chosen terms from the expansion sequence require this calculation for proper distribution.

Combinatorial mathematics simplifies the challenge of polynomial expansion by defining the number of ways each term can appear in the expanded expression.

Algebraic Expressions

Understanding algebraic expressions is critical when simplifying polynomials, especially during expansion. Here, algebraic expressions consist of terms involving variables and coefficients combined by addition, subtraction, multiplication, and division.
In dealing with \( \left(x^{2}+x+1\right)^{3} \), once expanded, you'll encounter mixed terms and various powers of variables. Simplification is necessary to combine like terms which share the same degree.

• First, identify which terms can be combined by checking their powers.
• Next, sum up coefficients of similar terms left after expansion.
• Finally, rearrange the expression to follow standard decreasing order by the degree of terms.

Algebraic expressions become more manageable with skillful manipulation and organization within binomial expansions. By breaking them down and then recombining them, we achieve a simpler, clearer representation of our mathematical outcomes.