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 is equal to zero:

Graph of Inequality

Factors

We can take a polynomial, such as:

f(x) = ax4 + bx3 + ...

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:

ax2 + 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( x2 − px − qx + pq )
= ax2 − a(p+q)x + apq

Now let us compare:
Quadratic: ax2 +bx +c
Expanded Factors: ax2 −a(p+q)x +apq

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

p+q = −b/a

And c = apq, so:

pq = c/a

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:

ax2 + 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

x2 − (sum of the roots)x + (product of the roots) = 0

So we end up with:

x2 − 10x + 23 = 0

And here is the plot:

Cubic

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

ax3 + bx2 + cx + d

As with the Quadratic, let us expand the factors:

a(x−p)(x−q)(x−r)
= ax3 − a(p+q+r)x2 + a(pq+pr+qr)x − a(pqr)

And we get:

Cubic: ax3 +bx2 +cx +d
Expanded Factors: ax3 −a(p+q+r)x2 +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 (for odd degree polynomials like cubics)