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:
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 , where is a factor of the constant term (44) and is a factor of the leading coefficient (1 in this case).
Step 1: List possible rational roots
The factors of are: So, the possible rational roots are .
Step 2: Test possible roots using synthetic division or direct substitution
Let's start by testing :
Substitute into the polynomial: So, is a root of the equation.
Step 3: Factor the polynomial
Since is a root, we can factor the polynomial as: We now perform synthetic division of by :
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.Ask a new question for Free
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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