Question

If \(\mathrm{x}=2+2^{2 / 3}+2^{1 / 3}\) then the value of \(\mathrm{x}^{3}-6 \mathrm{x}^{2}+6 \mathrm{x}\) is (a) 1 (b) 2 (c) 3 (d) 4

Short answer

Answer: (a) 1

Step-by-step solution

Substituting the given value of x

We will substitute the given value of x in the expression x^3 - 6x^2 + 6x: Expression = (2 + 2^(2/3) + 2^(1/3))^3 - 6(2 + 2^(2/3) + 2^(1/3))^2 + 6(2 + 2^(2/3) + 2^(1/3))

Cubing and squaring the expression

We need to cube the expression (2 + 2^(2/3) + 2^(1/3)) and square the expression (2 + 2^(2/3) + 2^(1/3)). To do this, we can apply the binomial theorem formula: (a + b + c)^3 = a^3 + b^3 + c^3 + 3a^2b + 3ab^2 + 3a^2c + 3ac^2 + 3b^2c + 3bc^2 + 6abc In this case, a = 2, b = 2^(2/3), and c = 2^(1/3). Proceed to calculate the different terms of this expansion. (a + b + c)^2 = a^2 + b^2 + c^2 + 2ab + 2ac + 2bc Again, a = 2, b = 2^(2/3), and c = 2^(1/3). Calculate the different terms of this expansion too.

Simplify the obtained terms

Now, we will plug in the computed values of the terms from the previous step and simplify the expression. As we do this, we will observe some terms that cancel out and simplify the resultant expression.

Obtain the final value

After simplifying the expression in the previous step, we will find that the final value of the expression x^3 - 6x^2 + 6x for the given value of x is: Answer: (a) 1

Key concepts

Binomial Theorem

The Binomial Theorem is a powerful algebraic tool for expanding expressions that have been raised to a power. It lets us handle expressions involving sums more easily.

• The theorem provides a formula: \((a+b)^n = a^n + \binom{n}{1} a^{n-1} b + \binom{n}{2} a^{n-2} b^2 + \ldots + b^n\), where \(\binom{n}{k}\) are binomial coefficients.
• In our problem, we've extended the theorem to three terms: \((a+b+c)^3\). This requires calculating various combinations, such as \(a^3, b^3,\) and cross terms like \(3ab^2\).
• Though complex, this method simplifies evaluating powers of binomials.

Understanding this helps in expanding polynomial expressions without manually multiplying each term, saving time and reducing errors.
The binomial formula also reveals the symmetry and patterns in polynomial expansions.

Cubing Expressions

Cubing expressions involves raising an algebraic expression to the power of three. This process builds on the concept from the Binomial Theorem when dealing with multiple terms.

• For an expression like \((a + b + c)^3\), each term must interact with every other term according to the pattern from binomial expansion.
• One expands the expression to include terms like \(a^3, b^3, c^3\), and multiple cross terms like \(3a^2b\).
• The expansion becomes a systematic way to handle polynomials, ensuring each interaction is accounted for.

Using this technique helps us break down complex cubic polynomials, making their analysis and simplification more manageable.
Knowing the cubed term interactions helps in understanding the structure of polynomials, critical for further simplifications.

Simplifying Algebraic Expressions

Simplifying algebraic expressions is about making them easier to work with by reducing their complexity. This often involves combining like terms, and applying known algebraic identities.

• In our problem, after expanding with the Binomial Theorem, we gather like terms and terms that cancel each other out.
• The simplification might reveal terms that subtract out completely, immensely reducing complexity.
• Being effective at simplification paves the way for solving equations more efficiently.

Such simplifications often make hidden patterns visible, leading to straightforward solutions.
This process is crucial for students as it provides the foundation for solving complex algebraic equations in a step-by-step manner.

Polynomial Evaluation

Polynomial evaluation is the process of finding the value of a polynomial expression for a given variable. In this exercise, we evaluate \(x^3 - 6x^2 + 6x\) for a specific value of \(x\).

• The expression for \(x\) in our problem was \(2 + 2^{2/3} + 2^{1/3}\), which was complex and required expansion and simplification before evaluation.
• During evaluation, keep track of each term and calculate its contribution to the total.
• Evaluation may often lead us through paths of simplification, especially when some terms cancel out, making the final value realizable.

This approach to polynomial evaluation helps in double-checking correctness and understanding result significance.
It's a critical skill for accurately computing results in algebraic tasks.