2.3. Relationship between Zeroes and Coefficients of a Polynomial

Let's begin our discussion on quadratic polynomials. Can anyone remind me what a quadratic polynomial looks like?

Is it in the form ax^2 + bx + c?

Exactly! Now, if we have a quadratic polynomial like p(x) = 2x^2 - 8x + 6, how do we find its zeroes?

We can factor it or use the quadratic formula.

Are the zeroes related to the coefficients?

Great question! The sum of the zeroes, α + β, is given by the formula -b/a, and the product, αβ, is given by c/a. Let's calculate these for our polynomial.

So for p(x), -(-8)/2 = 4 is the sum, and 6/2 = 3 is the product!

Yes, exactly! So the zeroes here, 1 and 3, satisfy both conditions. Let's summarize this concept.

"Remember: For any quadratic polynomial ax^2 + bx + c, the relationships between zeroes and coefficients are:

Let's move on to an example. Consider the polynomial p(x) = 3x^2 + 5x - 2. How can we find its zeroes?

We can factor it into (3x - 1)(x + 2) and set each factor to zero.

Correct! And what are the zeroes then?

The zeroes are 1/3 and -2.

Now let's check the relationships. What do we see for the sum and product?

Sum: 1/3 - 2 = -5/3, which equals -5/3 indeed, and product: (1/3)*(-2) = -2/3, which fits too!

Fantastic! Keep practicing these relationships, as they are critical in future sections.

Moving on, let’s talk about cubic polynomials represented as p(x) = ax^3 + bx^2 + cx + d. Do we have a similar relationship?

Yes! The sum of the zeroes is still related to the coefficients.

"Exactly! The relationships for zeroes α, β, γ become:

Before we wrap up, let's review! What can we say about zeroes in terms of graphical representations?

The zeroes represent x-coordinates where the graph intersects the x-axis.

Absolutely! This is critical for understanding polynomial behaviors. And how many zeroes can a cubic polynomial have?

At most three zeroes!

"Excellent! Lastly, let's remember the formulas for any quadratic polynomial relation: