Solving Polynomial Equations: Review
Solving Polynomial Equations: Review
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The goal of this webpage is to present resources for you to review a core topic in algebra: solving polynomial equations. For engineering mathematics, you need to be able to determine by hand the roots of any cubic (degree 3) polynomial you are given. This requires you to do three tasks:
Identify a root r of the given cubic polynomial.
Divide the polynomial by x - r to obtain a quadratic (degree 2) polynomial.
Determine the roots of a quadratic polynomial.
- Identifying a root
This is usually done by trial and error. In the above cubic polynomial, you can find a root by
Finding x = p and x = q where P(p) > 0 while P(q) < 0. In this case, you know that the root x = r must be between p and q.
If r is a whole number, and c is a whole number, then r divides c. This is not true if c or r is not a whole number! Usually the divisors of c make good places to start the root-guessing process.
E.g. for P(x) = x^3 + x^2 + x - 2, we have P(0) < 0 while P(1) > 0. This tells us a root is between 0 and 1. The second fact does not apply, because the root is not a whole number.
Note: the polynomials I give you will always have at least one whole number root.
2. Polynomial/"Synthetic" Division
2. Polynomial/"Synthetic" Division
Once you have a linear factor (x - r) of P(x), you must divide P(x) by (x - r) to obtain a quadratic polynomial. Polynomial division (sometimes called "synthetic division") is covered in Paul's Online Notes.
Note: as (x - r) is a factor of P(x), we are guaranteed to have no remainder.
3. Solving Quadratic Equations
3. Solving Quadratic Equations
Quadratic equations are solved by factoring or through the quadratic formula (the "-b formula"). This is also reviewed in Paul's Online Notes:
The roots may be real or complex, so ensure you are also up to date on complex numbers.
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