Math Lesson 11.1.6 - Finding the Zeroes of a Polynomial

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Welcome to our Math lesson on Finding the Zeroes of a Polynomial, this is the sixth lesson of our suite of math lessons covering the topic of The Definition of Monomials and Polynomials, you can find links to the other lessons within this tutorial and access additional Math learning resources below this lesson.

Finding the Zeroes of a Polynomial

Sometimes, it is very useful to know the values of the variables of a polynomial for which the polynomial becomes zero. These values are known as the zeroes of the polynomial. For example, when factoring, simplifying or dividing two polynomials, it is useful to know the zeroes, as the abovementioned operations are done easier.

Constant polynomials do not have zeroes, as these polynomials only contain numbers, so it is impossible to make them zero. On the other hand, linear polynomials have one zero at maximum. For example, P(x) = 4 cannot be zero, as it does not contain any variable. On the other hand, the zero of the polynomial

is 3, because 2 · 3 - 6 = 0.

The zeroes of a quadratic polynomial represent the roots of the corresponding quadratic equation. For example, the zeroes of the quadratic polynomial

are 1 and 4, as

and

Example 6

Find the zeroes of the following polynomial

1. P(x) = x³ - x
2. P(x) = x² - 4x + 3

Solution 6

1. We can factorise the polynomial to identify the possible zeroes. Thus,
   P(x) = x³ - x
   = x(x²- 1)
   = x ∙ (x - 1)(x + 1)
   The polynomial becomes zero if any of its three component factors is zero. Hence, the above polynomial has three zeroes:
   x₁ = 0
   x₂ - 1 = 0
   x₂ = 1
   and
   x₃ + 1 = 0
   x₃ = -1

It is very important therefore to know the degree of a polynomial as it gives the maximum possible zeroes it may have. In other words, a n^th degree polynomial may have n or less zero points. This means the corresponding equation when P(x) is taken as 0, has n roots at maximum.

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6. Check your calculations for Polynomials questions with our excellent Polynomials calculators which contain full equations and calculations clearly displayed line by line. See the Polynomials Calculators by iCalculator™ below.
7. Continuing learning polynomials - read our next math tutorial: Operations with Polynomials

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