Solving Polynomial Equations: Review
Solving Polynomial Equations: Review
Embedded Files
The goal of this webpage is to present resources for you to review a core topic in precalculus algebra: solving polynomial equations. For linear analysis I, you need to be able to determine by hand the roots of any cubic (degree 3) polynomial I give you. 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.
Page updated
Google Sites
Report abuse