1. Introduction: The Numbers Hiding Inside a Polynomial
You already know what a zero of a polynomial is. A zero is a value of x that makes the polynomial equal to 0.
Take the polynomial x^2 - 5x + 6. To find its zeroes, you factorise it: (x - 2)(x - 3). So the zeroes are 2 and 3, because putting x = 2 or x = 3 makes the whole thing 0.
Now look at the polynomial again: x^2 - 5x + 6. The numbers sitting in front — the 1, the -5, the 6 — are called coefficients. The coefficient of x^2 is 1, the coefficient of x is -5, and the constant term is 6.
Here is the question for this whole lesson. The zeroes are 2 and 3. The coefficients are 1, -5, 6. Is there a quiet connection between them?
Stop scrolling. Add the two zeroes: 2 + 3. Multiply the two zeroes: 2 x 3. Write both answers down before reading on.
You got 5 and 6. Now look back at the coefficients: -5 and 6. Notice anything? The sum 5 is the -5 with its sign flipped. The product 6 is the constant term exactly. That is not a coincidence — and by the end of this lesson you will know why it must happen every single time.
You can now see that the zeroes and the coefficients of a polynomial are talking to each other.