Zeroes of a Cubic Polynomial

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We have seen that cubic polynomials are of the form

\(p\left( x \right):a{x^3} + b{x^2} + cx + d,\,\,a \ne {\rm{0}}\)

Any such polynomial will have, in general, three zeroes. For example, \(p\left( x \right):{x^3} - 6{x^2} + 11x - 6\) has the following three zeroes (verify that these are indeed the zeroes of the polynomial): \(x = 1,\,\,2,\,\,3\).

The three zeroes of a cubic polynomial might all be equal. For example, consider \(p\left( x \right):{\left( {x - 1} \right)^3}\). This has the three zeroes \(x = 1,\,\,1,\,\,1,\) which happen to be identical.

Another case which is possible is that two of the zeroes are equal, and the third is different. For example, consider \(p\left( x \right):{\left( {x - 1} \right)^2}\left( {x - 2} \right)\). This has the three zeroes: \(x = 1,\,\,1,\,\,2\).

Will a cubic polynomial always have three real zeroes? The answer is no. Just as a quadratic polynomial does not always have real zeroes, a cubic polynomial may also not have all its zeroes as real. But there is a crucial difference. A cubic polynomial will always have at least one real zero. Thus, the following cases are possible for the zeroes of a cubic polynomial:

  1. All three zeroes might be real and distinct.

  2. All three zeroes might be real, and two of them might be equal.

  3. All three zeroes might be real and equal.

  4. One zero might be real and the other two non-real (complex).

The reasons behind these properties of zeroes will become clear later. For now, you should try to memorize these facts so that things will be easier for you when you reach the stage of studying them in more detail.

Example 1: Consider the following cubic polynomial:


Which of the following are zeroes of this polynomial?

(A) \( - 1\)                               (B) \(1\)

(C) \(2\)                                  (D) \(3\)

(E)  \(4\)

Solution: The correct options are (B), (C) and (D). Verify the same by substitution.

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