How to Factor 3rd Degree Polynomials
Polynomials are a fundamental concept in mathematics, and factoring them is an essential skill for students and professionals alike. Among the various types of polynomials, third degree polynomials, also known as cubic polynomials, can be particularly challenging to factor. In this article, we will explore different methods to factor 3rd degree polynomials, including the Rational Root Theorem, synthetic division, and the quadratic formula.
1. The Rational Root Theorem
The Rational Root Theorem is a useful tool for finding possible rational roots of a polynomial. It states that if a polynomial has a rational root, then that root must be of the form p/q, where p is a factor of the constant term and q is a factor of the leading coefficient. By testing these possible roots, we can often find one or more real roots of the polynomial.
For example, consider the cubic polynomial f(x) = x^3 – 5x^2 + 6x – 8. The constant term is -8, and the leading coefficient is 1. The factors of -8 are ±1, ±2, ±4, and ±8, while the factors of 1 are ±1. By testing these possible roots, we find that x = 2 is a root of the polynomial. We can then use synthetic division to factor out (x – 2) and reduce the cubic polynomial to a quadratic one.
2. Synthetic Division
Synthetic division is a method for dividing a polynomial by a linear factor (x – a). It is a simplified version of long division and can be used to factor polynomials by finding their roots. In the previous example, we used synthetic division to factor out (x – 2) from f(x).
To perform synthetic division, write the coefficients of the polynomial in a row, with the leading coefficient on the left. Write the root (a) below the leftmost coefficient. Bring down the first coefficient, multiply it by the root, and add the result to the next coefficient. Repeat this process until you reach the last coefficient. The last number in the row is the remainder, and the coefficients in the row represent the quotient polynomial.
In our example, the synthetic division would look like this:
“`
2 | 1 -5 6 -8
| 2 -2 2
—————-
| 1 -3 4 -10
“`
The quotient polynomial is x^2 – 3x + 4, which can be factored further using the quadratic formula or by completing the square.
3. The Quadratic Formula
The quadratic formula is a method for finding the roots of a quadratic polynomial (ax^2 + bx + c = 0). By using the quadratic formula, we can find the roots of the quotient polynomial from the previous example (x^2 – 3x + 4) and factor the cubic polynomial completely.
The quadratic formula is given by:
x = (-b ± √(b^2 – 4ac)) / (2a)
In our example, a = 1, b = -3, and c = 4. Plugging these values into the quadratic formula, we get:
x = (3 ± √(9 – 16)) / 2
x = (3 ± √(-7)) / 2
Since the discriminant (b^2 – 4ac) is negative, the roots are complex numbers. Therefore, the complete factorization of the cubic polynomial f(x) = x^3 – 5x^2 + 6x – 8 is:
f(x) = (x – 2)(x – (3 + √7)i)(x – (3 – √7)i)
In conclusion, factoring 3rd degree polynomials can be achieved through various methods, including the Rational Root Theorem, synthetic division, and the quadratic formula. By understanding and applying these techniques, you can successfully factor cubic polynomials and solve related problems in mathematics.
