How Do You Find the Sum and Product of the Roots of a Polynomial Equation?

Use Vieta's formulas: the sum and product of all roots, counted with multiplicity, come directly from the polynomial's coefficients. You do not need to solve the equation; the product's sign depends on whether the degree is even or odd.

For a quadratic

$$ax^2+bx+c=0,$$

with roots $\alpha$ and $\beta$, factorisation gives

$$a(x-\alpha)(x-\beta)=a\left[x^2-(\alpha+\beta)x+\alpha\beta\right].$$

Expanding and comparing coefficients with $ax^2+bx+c$ produces

$$\alpha+\beta=-\frac{b}{a}, \qquad \alpha\beta=\frac{c}{a}.$$

For example, the roots of $3x^2-7x-6=0$ have sum

$$\alpha+\beta=-\frac{-7}{3}=\frac{7}{3},$$

and product

$$\alpha\beta=\frac{-6}{3}=-2.$$

These results are obtained without factorising or using the quadratic formula.

For a polynomial of degree $n$,

$$a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0=0,$$

with roots $r_1,r_2,\ldots,r_n$, including repeated and complex roots, the general results are:

The full pattern comes from expanding $a_n(x-r_1)\cdots(x-r_n)$. Choosing one root term and factors of creates the coefficient of , so that coefficient is . Choosing every root term creates the constant term . This explains both formulas and the alternating sign.

Consider

$$2x^3-5x^2+7x-3=0.$$

The sum of its three roots is

$$-\frac{-5}{2}=\frac{5}{2},$$

while their product is

$$(-1)^3\frac{-3}{2}=\frac{3}{2}.$$

As a further check, for $x^4-2x^3+5x-8=0$, the missing $x^2$ term does not affect either requested result. The sum is , while the product is , because the degree is even. Notice that all four roots are included even if some are repeated or non-real. This also provides a quick check on roots found later by factorisation or technology.

A common misconception is that the product is always $a_0/a_n$. That is only true for even-degree polynomials; for odd degree, $(-1)^n$ reverses the sign. Another common error is overlooking a missing term: in , the coefficient of is zero, so the sum of the roots is zero.

Exam technique: This is IB Mathematics AA HL, sub-topic AHL 2.12 Polynomial functions, and can appear on Paper 1 or Paper 2. Show the coefficient substitution clearly, including the factor $(-1)^n$, and solve for individual roots only if explicitly required.