eli5 – What does root of an polynomial mean and what’s its importance?
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Let’s define a polynomial as a function f(x) = ax² + bx + c
Of course, you could have x³ or x^4 or other terms in there as well, but we’ll use a quadratic for simplicity.
The roots of a polynomial are the values of x such that f(x) = 0
For example the easiest one, f(x) = x², only has one root – x = 0. When x = 0, f(x) = 0 because (0)² = 0.
The importance of the roots depends on the kind of problem you are trying to solve, but in general it tells you where the function goes from positive to negative (in cases where it has multiple roots,) or that the function is only ever positive or negative (in cases where it either has no roots, or only has one root)
The roots of a polynomial are any values of the independent variable (typically x) that causes the value of the polynomial to be zero.
Example: y = x^2 has only one root, because only x=0 causes y=0.
Example: y=x^2 + 1 has no root because all real values of x cause y>0. (If you allow for imaginary numbers, it has 2 roots x=i and x=-i, where i = √-1).
Example: y=x^2 – 1 has two roots because x=1 and x=-1 both cause y=0.
This is important because methods of solving polynomials are easier if you’re solving for zero. For example, if you want to solve for what value of x yields some nonzero value of y, it is helpful to subtract that nonzero value from the polynomial before solving it.
the roots of a polynomial are the places it crosses the X-axis, that is to say, the X values for which the Y values equal 0.
every polynomial can be rewritten in terms of its roots, and this is called factoring. a factored polynomial is much easier to deal with, whether you’re trying to solve for a specific value or if you’re doing some more complicated analysis.
Something the answers have not touched on is WHY they are important. Many equations are just roots of polynomials if you rearrange them.
3x = x² – 2
could be an equation we want to solve. It can be rewritten as
x² – 3x – 2 = 0
Now we can think of x² – 3x – 2 as our polynomial and the roots are the solution to our equation. The advantage is that if we know how to calculate the roots of a polynomial, we can solve many equations by looking at it from the perspective of roots of polynomials.
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