Real Zeros of Polynomials

Instructor: Babita Kuruvilla

Babita has an electrical engineering degree and has taught engineering students and college students preparing for medical and dental college admissions tests.

In this lesson, we'll tackle polynomial equations and learn how to solve for their zero values. We'll also work through some sample problems using Descartes' Rule of Signs.

Mathematical Expressions

Let's say you're given ten words and asked to form a complete sentence about how you feel using some or all of those words. Maybe you formed the expression, 'I am happy it's a new day!'

Now, what if you're given some random numbers and letters and asked to write an expression in mathematical form? You'd combine those numbers and letters along with arithmetic operations and string them together to write a mathematical expression like 5ab - 2c + 4 = 0.

Polynomials

We can express a mathematical relationship with just one or multiple terms. When the expression has only one term, it is called a monomial; when it has two terms, it's called a binomial. A trinomial expression is a combination of three terms. These terms can be made up of a combination of numbers and variables and explain the relationships among them.

An expression composed of two or more terms that have variables in them is called a polynomial. Let's say that P(x) is a polynomial function. If we let P(x) = 0 and solve for the values of x, we get the zeros of the polynomial!

Zeros of a Polynomial

Let's break down the process of finding zeros of a polynomial into the basic steps.

Step 1): Write an expression based on what's given.

Step 2): Set the polynomial equal to zero.

Step 3): Solve for the variable(s) using polynomial factorization.

The degree of the polynomial tells us the maximum number of distinct zeros it will have. For example, x2 - 4 has a degree of 2 and therefore two distinct zeros.

Finding Zeros

What if we are given the expression x2 + 5x + 4? Notice that it has three terms, which makes it a trinomial. Can we solve for x? Sure we can, by finding the zeros of the polynomial. Since this polynomial has a degree of 2, we expect two distinct zeros.

Step 1): Write an expression based on what's given: x2 + 5x + 4.

Step 2): Set the polynomial equal to zero: x2 + 5x + 4 = 0.

Step 3): Solve for the variable(s) using polynomial factorization:

  • x2 + 5x + 4 = 0
  • (x + 1)(x + 4) = 0
  • (x + 1) = 0 and (x + 4) = 0
  • x = -1 and x = -4

As we predicted earlier, we get two distinct zeros (-1 and -4) for the polynomial.

Multiplicity of Zeros

Can you find the zeros of the polynomial P(x) = (x + 3) + (x - 1)2 + (x - 2)?

Step 1): P(x) = (x + 3)4 + (x - 1)2 + (x - 2).

Step 2): P(x) = (x + 3)4 + (x - 1)2 + (x - 2) = 0.

Step 3): (x + 3)(x + 3)(x + 3)(x + 3) + (x - 1)(x - 1) + (x - 2) = 0.

  • (x + 3) = 0; (x - 1) = 0; (x - 2) = 0
  • x = - 3, 1, 2

Notice that P(x) has the zero '-3' four times, the zero '1' twice, and the zero '2' once. So, we can say that the multiplicities of the zeros for this polynomial are: four for -3, two for 1, and one for 2.

Descartes' Rule of Signs

Whenever we solve for real zeros, it's good to know ahead of time how many positive real zeros and negative real zeros we can expect. Rene Descartes, a French philosopher and mathematician, theorized that for a polynomial P(x) with real coefficients, the number of real positive and negative zeros can be found by following certain rules, which came to be known as Descartes' Rule of Signs.

Number of Positive Real Zeros

According to Descartes' Rule of Signs, the number of positive real zeros within a polynomial P(x) is equal to the number of changes in sign or an even number subtracted from it.

For example, the polynomial P(x) = x4 + x3 - x2 + x - 2 has three sign changes:

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