# Polynomials: Sums and Products of Roots

I want to share with you something interesting about the roots of polynomials ...

- What happens when you
**add**the roots? - What happens when you
**multiply**the roots?

## Roots of a Polynomial

A "root" (or "zero") is where the polynomial |

## Factors

You can take a polynomial, such as:

f(x) = ax^{4} + bx^{3} + ...

And then factor it like this:

f(x) = a(x-p)(x-q)(x-r)...

Then p, q, r, etc are the **roots** (where the polynomial equals zero)

## Quadratic

Let's try this with a Quadratic:

ax^{2} + bx + c

When the roots are p and q, the same quadratic becomes:

a(x-p)(x-q)

*Is there a relationship between p,q and a,b,c?*

Let's expand it, by multiplying **(x-p)** by **(x-q)**:

a(x-p)(x-q)

= a( x^{2} - px - qx + pq )

= ax^{2} - a(p+q)x + apq

Quadratic: | ax^{2} |
+ bx | + c |

Expanded Factors: | ax^{2} |
- a(p+q)x | + apq |

We can now see that bx = - a(p+q)x, so:

And c = apq, so:

And we get this simple result:

- Adding the roots gives
**-b/a** - Multiplying the roots gives
**c/a**

This can help us answer questions.

### Example: What is the equation whose roots are 5 + √2 and 5 - √2

The sum of the roots is (5 + √2) + (5 - √2) = **10**

The product of the roots is (5 + √2) (5 - √2) = 25 - 2 = **23**

And we want an equation like:

ax^{2} + bx + c = 0

When **a=1** we can work out that:

- Sum of the roots =
**-b/a**=**-b** - Product of the roots =
**c/a**=**c**

Which gives us this interesting result

x^{2} - (sum of the roots)x + (product of the roots) = 0

So we end up with:

x^{2} - 10x + 23 = 0

And here is the plot:

## Cubic

Now let us look at a Cubic (one degree higher than Quadratic):

ax^{3} + bx^{2} + cx + d

As with the Quadratic, let us expand the factors:

a(x-p)(x-q)(x-r)

= ax^{3} - a(p+q+r)x^{2} + a(pq+pr+qr)x - a(pqr)

And we get:

Cubic: | ax^{3} |
+ bx^{2} |
+ cx | + d |

Expanded Factors: | ax^{3} |
- a(p+q+r)x^{2} |
+ a(pq+pr+qr)x | - apqr |

Hey! We get the same sort of thing:

- Adding the roots gives
**-b/a**(exactly the same as the Quadratic) - Multiplying the roots gives
**-d/a**(similar to +c/a for the Quadratic)

(We also get pq+pr+qr = c/a, which can itself be useful.)

## Higher Polynomials

The same pattern continues with higher polynomials.

In General:

- Adding the roots gives
**-b/a** - Multiplying the roots gives (where "z" is the constant at the end):
**z/a**(for even degree polynomials like quadratics)**-z/a**