Meaning of polynomial roots

Arnav Zek
2 min readMay 4, 2019


2x² + 3x + 5, if it’s a line it will be Y = 2x² + 3x + 5

so the roots will be value of x that makes y = 0 or in other words the point on that line to fall on the x axis

if it’s a financial model roots will be the values that makes the output zero

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how do we find one?

we factor it , in a more expressive way backfoil it.

but to understand that better we need to observe

here’s a dumy factor: (x+c)(x+ (c+n) ), where n can be positive or negative constant

four parts will be outputted from it: x² + (c+n)x + cx + c(c+n), notable points :-

  • the last part is the only constant we have. so we have our first rule of backfoiling that multiplying the bracket constant should give our original constant
  • our degree one buddies in the middle are going eat each other. so our second rule is that single degree parts should add up to give our original single degree part

with these two notions in place we can find factor of any two degree polynomial, in one line: find the factors of the constant which on addition will give our degree one part and naturally on multiplication will give out constant part

finding these rules for degree 3 could be a little tricky, you can think the same way we turned degree 2 to two degree one factors we can change degree 3 to a degree 2 factor and then to degree one factor which will be our roots

it will be of format (x² + x¹ + 2 )(x¹ +1) => x³ + x² + x² + x + 2x + 2

so, we have the format of factors of our degree 3 polynomials. can you find the rules to find the factors

here we go:

  • first rule stays the same: multiplying constant should give the original constant
  • there will be single degree 3 part so, the same rule for that

*gif needed*

  • coefficient of degree 2 (in the factor format) + { coefficient of degree 1 (in the first bracket) * coefficient of degree 1 (in the second bracket) } should give the coefficient of the degree 2 part in the polynomial
  • in the three part bracket coefficient of first degree x + constant should give the degree 1 part of the polynomial