Using the Multiple Roots Theorem to Solve a Polynomial Equation

In this video, we cover the multiple roots theorem of polynomials, which basically states that if a is a root of p(x) = 0 with multiplicity 3, then a is also a root of p'(x) with multiplicity 2 and a root of p''(x) with multiplicity 1.
This then extends to factors where if (x - a)^3 is a factor of p(x), then (x - a)^2 is a factor of p'(x) and (x - a) is a factor of p''(x).
We use this theorem to find all of the roots of the equation p(x) = x^4 - 6x^3 + 12x^2 - 10x + 3 = 0
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[@MultiRedskins12]

Why do long division? The polynomial is a monic. The product of the roots is 3. A triple root at x=1 means the remaining root is 3. Much quicker!

[@mhonthungovung7122]

Thank you so much for this video , it really helps me a lot ..very clear voice and well understandable too🤗

[@MasterWuMathematics]

You're welcome 😊

[@subhamkarmakar356]

Sir what if we have more than one multiple roots ?

[@MasterWuMathematics]

This theorem should still be valid. Do you have an example?

[@subhamkarmakar356]

@@MasterWuMathematics oh I just did some problems, and realised a pattern though that if there are k multiple roots then we should have the gcd polynomial of degree k ig.

[@zethayn]

Thank you!