Question

What is the value of \(x\) ? (1) \(x^3-3 x^2+3 x=1\) (2) \((x-1)^2=0\) A. 1 alone, not 2 alone B. 2 alone, not 1 alone C. 1 and 2 together (need both) D. 1 alone or 2 alone E. 1 and 2 together are not sufficient

Short answer

D. 1 alone or 2 alone

Step-by-step solution

Understanding the Equation (1)

Start with the first equation: \(x^3 - 3x^2 + 3x = 1\). Notice that this is a polynomial equation in terms of \(x\). We should factorize or simplify it to find the value of \(x\).

Simplify Equation (1)

Rearrange the equation to set it to zero: \(x^3 - 3x^2 + 3x - 1 = 0\). Check if there are any obvious roots by substituting small integer values for \(x\).

Solve for \(x\) from Equation (1)

Notice that substituting \(x = 1\) into the polynomial equation: \(1^3 - 3(1)^2 + 3(1) - 1\) results in \(1 - 3 + 3 - 1 = 0\), so \(x = 1\) is a solution to the first equation.

Analyze Equation (2)

Now consider the second equation: \((x-1)^2 = 0\). Simplify it to find the value of \(x\).

Solve for \(x\) from Equation (2)

Notice that \((x-1)^2 = 0\) implies \(x - 1 = 0\). Therefore, \(x = 1\) is the solution for the second equation.

Final Analysis

Both equations independently lead to the solution \(x = 1\). So, either equation alone is sufficient to determine the value of \(x\).

Key concepts

Polynomial equations

Polynomial equations consist of terms including variables raised to whole number exponents combined using addition, subtraction, and multiplication. The general form of a polynomial equation is: ax^n + bx^(n-1) + ... + k = 0.In the given exercise, the polynomial equation is: x^3 - 3x^2 + 3x = 1. The polynomial degree is determined by the highest exponent, which in this case is 3. This makes it a third-degree polynomial (a cubic polynomial). Understanding polynomial equations is crucial for solving complex algebraic problems.

Next, let’s focus on how to manipulate this polynomial.

Factoring

Factoring is a method of simplifying polynomial equations by expressing them as a product of simpler polynomials. In the given exercise, we deal with the polynomial x^3 - 3x^2 + 3x - 1. To factor or simplify it, we rearrange it to get x^3 - 3x^2 + 3x - 1 = 0.

A powerful approach is to use known roots or trial and error with small integers to find factors. In this case, substituting x = 1 into the polynomial equation simplifies the equation to: 1 - 3 + 3 - 1 = 0, revealing that x = 1 is a root of the polynomial.

This allows us to factorize the polynomial's components, simplifying our calculations.

Substitution method

The substitution method involves replacing a variable with a value or another expression to simplify an equation. In our problem, we used substitution to solve for x. Specifically, we evaluated x = 1 in the polynomial to verify if it satisfies the equation. When we substituted x = 1 into x^3 - 3x^2 + 3x - 1, the equation did indeed hold true.

This method helps to quickly identify whether a potential solution is correct and is often used in conjunction with factoring to break down complex polynomials into simpler, solvable parts.

Equation analysis

Equation analysis is the process of dissecting an equation to understand its components and solve it accurately. For the first equation, x^3 - 3x^2 + 3x = 1, analyzing it involved rearranging and substituting values to discover the solution. Similarly, for the second equation, (x-1)^2 = 0, we analyzed it by recognizing that squaring function and its properties.

The simplest understanding of (x-1)^2 = 0 simplifies directly to x - 1 = 0, indicating x = 1. This shows the importance of analyzing equations thoroughly to identify the correct approach for solutions, whether through simplification, factoring, or substitution methods.