Math Problem Statement
The function f, of, x, equals, x, cubed, plus, 2, x, squared, minus, 10, x, minus, 20f(x)=x3+2x2−10x−20 has at least one integer root. Use the integer root theorem to find that root, then proceed to find all complex roots. (Note: roots may be integer, rational, irrational, and/or complex.)
Solution
To solve the given problem, we will follow these steps:
Step 1: Apply the Integer Root Theorem
The integer root theorem states that any possible rational root of a polynomial must be a factor of the constant term divided by a factor of the leading coefficient.
The given polynomial is:
- The constant term is .
- The leading coefficient (of ) is .
Thus, the possible integer roots are the factors of . The factors of are:
Step 2: Test Possible Integer Roots
Now, we'll substitute these possible integer roots into and see which one gives .
- For :
- For :
- For :
- For :
So, is an integer root.
Step 3: Perform Polynomial Division
Now that we have as a root, we can factor as . To find the quotient, we perform synthetic division of by .
Synthetic Division:
We divide by .
-2 | 1 2 -10 -20
| -2 0 20
---------------------
1 0 -10 0
The quotient is , and the remainder is 0. Therefore, the polynomial can be factored as:
Step 4: Solve for the Remaining Roots
Now, we solve the quadratic for the remaining roots.
Thus, the roots are and .
Final Answer:
The roots of the polynomial are:
- (integer root),
- (irrational root),
- (irrational root).
Would you like more details on any of the steps or further explanation?
Related Questions:
- How do you perform synthetic division for higher-degree polynomials?
- Can the integer root theorem be applied to any polynomial?
- What are the conditions for a polynomial to have complex roots?
- How do we check for rational roots in a polynomial using the Rational Root Theorem?
- What is the geometric significance of irrational roots on the graph of a polynomial?
Tip:
When solving for roots of polynomials, always check for integer roots first, as they simplify the process of factorization and help you find more complex roots later.
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Math Problem Analysis
Mathematical Concepts
Algebra
Polynomials
Roots of Polynomials
Formulas
f(x) = x^3 + 2x^2 - 10x - 20
Synthetic division
Quadratic equation for solving x^2 - 10 = 0
Theorems
Integer Root Theorem
Quadratic Root Theorem
Suitable Grade Level
Grades 10-12
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