Polynomials: Zeroes and Coefficients

The relationship

For a quadratic polynomial $ax^2 + bx + c$ with zeroes $\alpha$ and $\beta$:

Worked example: finding zeroes and verifying the relationship

Find the zeroes of $6x^2 - 7x - 3$ and verify the relationship between the zeroes and the coefficients.

Solution: Using the quadratic formula with $a=6, b=-7, c=-3$:

So $x = \dfrac{3}{2}$ or $x = -\dfrac{1}{3}$. Checking against the coefficients:

Both match, so the zeroes are verified.

Worked example: building a polynomial from sum and product

Find a quadratic polynomial whose zeroes have sum $\sqrt{2}$ and product $\dfrac{1}{3}$.

Solution: A quadratic with sum $S$ and product $P$ of zeroes is $x^2 - Sx + P$, so here:

Multiplying through by 3 to clear the fraction gives an equivalent polynomial, $3x^2 - 3\sqrt{2}x + 1$, any nonzero multiple of a valid answer is also valid.