Math Problem Statement

Find all solutions of the following polynomial equation.

x^3−11x−4x^2+44=0

Solution

Let's start by rewriting the given polynomial equation in standard form:

x34x211x+44=0x^3 - 4x^2 - 11x + 44 = 0

Now, we will attempt to find the solutions by factoring or using the Rational Root Theorem, which suggests that any rational root of the equation will be of the form ±pq\pm \frac{p}{q}, where pp is a factor of the constant term (44) and qq is a factor of the leading coefficient (1 in this case).

Step 1: List possible rational roots

The factors of 4444 are: ±1,±2,±4,±11,±22,±44\pm 1, \pm 2, \pm 4, \pm 11, \pm 22, \pm 44 So, the possible rational roots are ±1,±2,±4,±11,±22,±44\pm 1, \pm 2, \pm 4, \pm 11, \pm 22, \pm 44.

Step 2: Test possible roots using synthetic division or direct substitution

Let's start by testing x=2x = 2:

Substitute x=2x = 2 into the polynomial: 234(22)11(2)+44=81622+44=02^3 - 4(2^2) - 11(2) + 44 = 8 - 16 - 22 + 44 = 0 So, x=2x = 2 is a root of the equation.

Step 3: Factor the polynomial

Since x=2x = 2 is a root, we can factor the polynomial as: (x2)(quotient)=0(x - 2)(\text{quotient}) = 0 We now perform synthetic division of x34x211x+44x^3 - 4x^2 - 11x + 44 by x2x - 2:

2 & 1 & -4 & -11 & 44 \\ & & 2 & -4 & -30 \\ \hline & 1 & -2 & -15 & 0 \\ \end{array}$$ So the quotient is $$x^2 - 2x - 15$$. Therefore, the factored form of the polynomial is: $$(x - 2)(x^2 - 2x - 15) = 0$$ ### Step 4: Solve the quadratic equation Now, solve the quadratic $$x^2 - 2x - 15 = 0$$ using the quadratic formula: $$x = \frac{-(-2) \pm \sqrt{(-2)^2 - 4(1)(-15)}}{2(1)}$$ $$x = \frac{2 \pm \sqrt{4 + 60}}{2} = \frac{2 \pm \sqrt{64}}{2} = \frac{2 \pm 8}{2}$$ So, the two solutions are: $$x = \frac{2 + 8}{2} = 5 \quad \text{and} \quad x = \frac{2 - 8}{2} = -3$$ ### Step 5: Final solutions The solutions to the equation are: $$x = 2, \quad x = 5, \quad x = -3$$ Do you need any further details or have questions about the solution? ### Related Questions: 1. How can we verify if all solutions are correct for a cubic equation? 2. How does synthetic division help in solving polynomials? 3. What is the Rational Root Theorem, and how is it applied in factoring? 4. Can cubic equations always be factored using simple methods? 5. What are the limitations of using the Rational Root Theorem? **Tip:** Always verify roots by substituting them back into the original equation to ensure correctness.

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Math Problem Analysis

Mathematical Concepts

Algebra
Polynomial Equations
Factoring
Rational Root Theorem

Formulas

Rational Root Theorem: Possible roots = ±p/q where p is a factor of the constant term and q is a factor of the leading coefficient
Quadratic Formula: x = (-b ± √(b² - 4ac)) / 2a

Theorems

Rational Root Theorem
Fundamental Theorem of Algebra

Suitable Grade Level

Grades 10-12