First of all let's start with a lttle explanationn of waht a polynomial is.

A polynomial is a function of x denoted f(x) which is defined as

f(x)=a_{0}+a_{1}x+. . .+a_{n}x^{n}

which in other words is a sum of x's raised to some positive power multiplied by some number a_{i} which is called the coefficent

Examples of polynomials:

f(x)=mx+c -> linear function

f(x)=ax^{2}+bx+c->quadratic

f(x)=ax^{3}+bx^{2}+cx+d->cubic

Now, what does it mean to solve a polynomial? It simply means for which values of x we have that f(x)=0, where f is our polynomial. Graphically, in x-y coordniate system if y=f(x) where does the curve cross the x-axis.

How does it work

Case1: Linear function

for f(x)=mx+c we set f(x)=0 in the equation and solve for x.

Case 2:quadratic

f(x)=ax^{2}+bx+c

Before solving the equation we have to check whether there are any.

When we draw f(x) on the x-y plane we we see that ot represents a parabola.

(Normaly here we would draw the picture)

We can also observe that the curve has a lowest point. As we cab see if the lowest point is below the x-axis we have two solutions, has one solution if it touches the axis, or no solutions if it is above the x-axis. Now the question is how to check it without having to draw the graph. What we do, we calculate the discriminant. Notice that the lowest point of the parabola is at x=b/2a (Shown later on by introducing derivatives). When we plug this value into our equation we get

y=a(b/2a)^{2}+b(b/2a)+c

but we know that it can be positive, zero or negative, therefore if y>0 at this point we said earlier that the polynomial has two solutions so

y>0 means 02+b(b/2a)+c

rewritting it slightly we get b^{2}-4ac > 0. Similarly this equals zero or less that zero if it has one or no solutions respectively. Therefore we found the expression for the number of solutions in terms of coefficients. b^{2}+4ac is called the discriminant.

We now want to find the particular solutions of quadratic equation. There are several ways of doing this. The quickest one is to factorise it by using little trick. Observe that we can rewrite quadratic which has solutions p,q in the following form:

y=m(x-p)(x-q)

Why? Well, if we plug in the values of p and q we get y=0 an remember that polynomials are unique up to value of m. For simplicity let m=1

y=(x-p)(x-q)=x^{2}-(p+q)x+pq

so, all we need to do is to express the coefficent of x in terms of p and q and we are done.

Case 3: cubic

For any cubic polynomial there is always at least one solution, the reason being that x^{3 }is a dominant term and so for large enough positive or negative values the function is above and below x-axis at some point (Picture should be added to visualize the result). There is clear procedure how to solve cubic polynomials however if we can spot one of them we are almost done as we only need to solve the remaining factor which is a quadratic equation and this we know how to solve.

Example:

f(x)=x^{3}-2x^{2}-x+2 can find x=1 as one of the solutions -> f(x)=(x-1)(x^{2}-x-2) -> f(x)=(x-1)(x+1)(x-2)