How to Factorize Complex Expressions Using Polynomial Division and Rational Root Theorem
Polynomial factorization is a fundamental technique in algebra, often encountered in mathematics and engineering. This process involves breaking down a polynomial into factors, which can be simpler polynomials or integers. In this guide, we will walk through the steps to factorize the complex expression (x^4 - 16x^3 86x^2 - 176x 40 - 65).
Understanding the Expression
Let's start by simplifying our given expression:
Expression: (x^4 - 16x^3 86x^2 - 176x 40 - 65)
First, combine the constant terms:
Expression: (x^4 - 16x^3 86x^2 - 176x - 25)
Using the Rational Root Theorem
The Rational Root Theorem helps identify possible rational roots of a polynomial. For the monic polynomial (x^4 - 16x^3 86x^2 - 176x - 25), the Rational Root Theorem tells us that any rational root is a factor of the constant term (-25) divided by the leading coefficient (1).
Step 1: Identify Rational Roots
Possible rational roots are the factors of -25: (±1, ±5, ±25).
Step 2: Testing Potential Roots
Using synthetic division or polynomial long division, we test these potential roots:
Testing (x 2): It is a root since the division results in a remainder of 0.Divide the polynomial by (x - 2):
Quotient: (x^3 - 14x^2 58x - 20)
Further Factoring Using Rational Root Theorem
Next, we factor the cubic polynomial (x^3 - 14x^2 58x - 20) using the Rational Root Theorem:
Testing (x 10): It is also a root since the division results in a remainder of 0.Divide the cubic polynomial by (x - 10):
Quotient: (x^2 - 4x 2)
Factoring the Quadratic Polynomial
Finally, we need to factor the quadratic polynomial (x^2 - 4x 2). We can use the quadratic formula: [x frac{4 pm sqrt{(-4)^2 - 4 cdot 1 cdot 2}}{2 cdot 1} frac{4 pm sqrt{16 - 8}}{2} frac{4 pm sqrt{8}}{2} 2 pm sqrt{2}]
Thus, the quadratic can be factored as:
[x^2 - 4x 2 (x - (2 sqrt{2}))(x - (2 - sqrt{2}))]Complete Factorization
The complete factorization of the original expression is:
[x^4 - 16x^3 86x^2 - 176x - 25 (x - 2)(x - 10)(x - (2 sqrt{2}))(x - (2 - sqrt{2}))]Substituting back, the final factorization is:
[x - 2x - 1 - 2sqrt{2}x - 2 - sqrt{2}]Conclusion
By systematically using the Rational Root Theorem and polynomial division, we can factorize complex expressions. This method is particularly useful for higher-degree polynomials where direct factorization might be challenging.