Vieta's Formulas and Symmetric Root Functions

Vieta's Formulas for Quadratic Equations

At its core, Vieta's formulas provide a direct relationship between the coefficients of a polynomial and the sums and products of its roots. For a general quadratic equation $a{x}^2+bx+c=0$ with roots $\alpha$ and $\beta$, the formulas are straightforward.

To get started, let's review these fundamental rules and see them in action.

The image below gives a quick visual summary of the formulas for both quadratic and higher-degree polynomials. We'll explore the generalized version later in the lesson.

Working with Symmetric Functions of Roots

A function of the roots, like $f(\alpha ,\beta )$, is symmetric if its value remains unchanged when the roots are interchanged. For example, ${\alpha }^2+{\beta }^2$ is symmetric because it's the same as ${\beta }^2+{\alpha }^2$. However, $\alpha -\beta$ is not symmetric.

The key to solving these problems is to express the given symmetric function in terms of the elementary symmetric polynomials: $\alpha +\beta$ and $\alpha \beta$. You've already seen this in the previous reading with ${\alpha }^2+{\beta }^2=(\alpha +\beta {)}^2-2\alpha \beta$.

Let's watch a video that works through another common symmetric function, ${\alpha }^3+{\beta }^3$.

This technique of using algebraic identities is fundamental. Here are the most common identities you'll need, which are also listed in the "JEE Prep Guide" resource you looked at earlier.

Extending to Higher-Degree Polynomials

Vieta's formulas are not limited to quadratics. They can be generalized for any polynomial of degree $n$. For a polynomial $P(x)=a_n{x}^n+a_{n-1}{x}^{n-1}+\cdots +a_{1}x+a_0$ with roots $r_1,r_2,\dots ,r_n$, the formulas follow a distinct pattern:

• Sum of the roots (taken one at a time): $\sum r_i=-\frac{a_{n-1}}{a_n}$
• Sum of the products of the roots (taken two at a time): ${\sum }_{i<j}{r}_i{r}_j=+\frac{a_{n-2}}{a_n}$
• Sum of the products of the roots (taken three at a time): ${\sum }_{i<j<k}{r}_i{r}_j{r}_k=-\frac{a_{n-3}}{a_n}$
• ...
• Product of all roots: $r_1{r}_2\dots r_n=(-1{)}^n\frac{a_0}{a_n}$

The sign alternates, starting with a negative. Let's see how this pattern works for cubic and quartic equations.

Now, let's see how these generalized formulas can be used with a clever identity to solve a problem that would otherwise be very difficult.

JEE Advanced Problem-Solving Patterns

In the JEE Advanced, you'll often encounter problems where the expression to be evaluated involves powers of roots in a sequence. A common and powerful technique is to use the fact that the roots themselves must satisfy the original equation.

Let's tackle a typical JEE Advanced problem.

This method of creating a recurrence relation is extremely useful.

For a final challenge, it's valuable to see how Vieta's formulas can be integrated with other mathematical concepts, such as infinite geometric progressions. The following video discusses a problem of this type from a past JEE Advanced exam. You don't need to work through the entire problem now, but watching the setup will give you a sense of the level of integration you should aim for.

Practice Problems

To solidify your understanding, try solving these problems from the provided resource. They cover different scenarios where Vieta's formulas are applicable.

Conclusion

In this lesson, you've learned how to apply Vieta's formulas to solve problems involving symmetric functions of roots, a cornerstone of polynomial theory for the JEE.

Key takeaways from our session:

• For a quadratic equation $a{x}^2+bx+c=0$, the sum of roots $\alpha +\beta =-b/a$ and the product $\alpha \beta =c/a$.
• Complex symmetric expressions can be evaluated by rewriting them in terms of $\alpha +\beta$ and $\alpha \beta$ using algebraic identities.
• Vieta's formulas extend to higher-degree polynomials with a predictable, alternating sign pattern.
• A powerful problem-solving technique involves substituting the root back into the polynomial equation itself to simplify expressions, often creating a recurrence relation.

In the next lesson, we will move to a different aspect of quadratic equations: solving inequalities using the wavy curve method. This graphical technique is essential for determining the intervals where a polynomial or rational function is positive or negative.