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Finding the factors and roots of a polynomial equation

Updated on June 23, 2011

What do we mean by "polynomial?"

Polynomial expressions are ones that have different powers of the same variable in, for example quadratics (or trinomials to Americans?) have an x2 term, an x term and a constant - cubics have an x3, an x2, an x and a constant. And so on...

The roots of a polynomial expression f(x) are where the graph of y = f(x) crosses the x-axis, or the values of x that make the expression equal to zero.

The simplest case of this would be factorising a quadratic equation in order to solve it, but the principles apply to all polynomials.

Factorising cubics

Cubic expressions, having x3 as the largest power of x, can be thought of as the product of a quadratic and a linear expression:

(x + n)(ax2 + bx + c) would give a cubic expression as the highest possible power of x would be given by multiplying the x term in the first bracket by the x2 term in the second.

You would generally be given the linear expression in the question, so the question becomes: how do we find the values of a, b, and c in the quadratic?

My method - click to enlarge
My method - click to enlarge

Comparing coefficients

If we know the linear expression that is a factor of our cubic, and the cubic expression we are trying to find the roots of, we can simply expand the brackets above and compare our coefficients - for example:

"Given that (x + 1) is a factor of f(x) = x3 + 4x2 - 15x - 18, solve f(x) = 0"

We multiply our (x + 1) by a "dummy" quadratic ax2 + bx + c to see how many of each power of x we get - we then compare the coefficients to f(x) in order to find the values of a, b, and c in our quadratic. See image!

Personally, I find it easier to multiply through by each term in the linear factor separately, and write the x3, x2, x and constant terms together - but that's just me.

We can then factorise the quadratic normally to find the missing roots.

In this example, the quadratic we need would be x2 + 3x - 18 (given the values of a, b, and c in the image above) which factorises to (x - 3)(x + 6), so our three factors are (x +1)(x - 3)(x + 6).

Our roots are whatever x values make f(x) = 0, which we can do by making each bracket zero - just like solving a quadratic.  This gives x = -1, x = 3, and x = -6.  This is where the graph of y = f(x) crosses the x-axis, if you want to check!

What if you're not given the linear factor?

Any linear factor of a polynomial can be found using the Factor Theorem, which is really just a special case of the Remainder Theorem - but that's another story!

Alternative method to comparing coefficients:


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    • Mr Homer profile imageAUTHOR

      Mr Homer 

      8 years ago from Yorkshire, England

      Sorry ;-)

      GCSEs are the final stage of compulsory education, for 14-16 year olds, and A Levels are for students up to 18 who then want to go to university - I don't know what these are equivalent to anywhere else?

      Thanks for the comment!

    • nadp profile image


      8 years ago from WEST PALM BEACH, FLORIDA

      You beat me to it, answering this question! I also teach math. You mentioned in your profile that you are using hubpages to build a gcse and math revision site. I'm not familiar with that terminology - I assume it's a British thing? What does it mean?


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