Question

What is the value of \(x\) ? (1) \(x^3-6 x^2+11 x=6\) (2) \(x>2\) 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

C. 1 and 2 together (need both)

Step-by-step solution

- Understand the Equation

Given equation: \[ x^3 - 6x^2 + 11x = 6 \] To find the value of \(x\), we need to solve this cubic equation.

- Solve the Cubic Equation

Rearrange the equation to \[ x^3 - 6x^2 + 11x - 6 = 0 \] Next, use the Rational Root Theorem to find possible rational roots. The possible rational roots are the divisors of the constant term \(-6\): \(\pm 1, \pm 2, \pm 3, \pm 6\).

- Test Potential Roots

Test the possible rational roots: For \(x = 1\): \[ 1^3 - 6(1)^2 + 11(1) - 6 = 1 - 6 + 11 - 6 = 0 \] Since \(x = 1\) satisfies the equation, it is a root.

- Derive Other Roots

Given that \(x = 1\) is a root, factorize the polynomial: \[ x^3 - 6x^2 + 11x - 6 = (x-1)(x^2 - 5x + 6) \] To get the remaining roots, solve the quadratic equation: \[ x^2 - 5x + 6 = 0 \] Using the quadratic formula: \[ x = \frac{5 \pm \sqrt{25 - 24}}{2} = 3 \text{ or } 2 \]

- Apply the Additional Condition

From statement (2), \( x > 2 \). Considering the roots, \( x = 1, x = 2, x = 3 \), the only value of \( x > 2 \) is \( x = 3 \).

- Evaluate Statements

Individually, statement (1) provides multiple roots (1, 2, 3) but does not single out one solution. Statement (2) alone (\( x > 2 \)) does not specify a unique value. Hence, both statements together are needed to conclude \( x = 3 \).

Key concepts

Cubic Equations

A cubic equation is a polynomial equation of the form \(ax^3 + bx^2 + cx + d = 0\). It has a degree of three. Solving cubic equations involves finding the roots (values of \(x\) that satisfy the equation). These roots can be real or complex numbers, and there can be up to three real roots. In our exercise, the cubic equation is given by \[ x^3 - 6x^2 + 11x = 6 \]. We first rearrange it into the standard form: \[ x^3 - 6x^2 + 11x - 6 = 0 \].

Rational Root Theorem

The Rational Root Theorem is a useful tool for solving polynomial equations. It states that any potential rational root of the polynomial \(ax^n + bx^{n-1} + \ldots + k = 0\) must be a fraction \(\frac{p}{q} \), where \(p\) is a factor of the constant term \(k\) and \(q\) is a factor of the leading coefficient \(a\). For our cubic equation \[ x^3 - 6x^2 + 11x - 6 = 0 \], the possible rational roots are the divisors of \(-6\): \(\pm 1, \pm 2, \pm 3, \pm 6\).

Quadratic Equations

A quadratic equation is a polynomial equation of degree two, typically written as \(ax^2 + bx + c = 0\). Quadratic equations can be solved using various methods, including factoring, completing the square, or the quadratic formula. In our case, after identifying \(x = 1\) as a root, we factorize the cubic equation into \[ (x-1)(x^2 - 5x + 6) = 0 \]. To find the remaining roots, we solve the quadratic equation \[ x^2 - 5x + 6 = 0 \], using the quadratic formula: \[ x = \frac{b \pm \sqrt{b^2 - 4ac}}{2a} \]. Substituting the values, we get \[ x = 3 \text{ or } x = 2 \].

Inequality Conditions

An inequality is a mathematical statement that compares two values or expressions using inequality symbols (>, <, \leq, \geq). In our problem, the additional condition given is \( x > 2 \). This condition narrows down the possible values of \(x\) from \(1, 2, \text{ and } 3\) to only \(x = 3 \). This is because \(x = 3 \) is the only value greater than 2 among the identified roots.

Polynomial Factorization

Polynomial factorization is the process of expressing a polynomial as a product of its factors. Factoring polynomials can simplify solving polynomial equations. When we identified \(x = 1\) as a root of \[ x^3 - 6x^2 + 11x - 6 = 0 \], it enabled us to factorize the polynomial: \[ (x-1)(x^2 - 5x + 6) = 0 \]. This factorization helps us solve the equation more easily by reducing it to the product of simpler equations, allowing us to find all roots.