Applications of Vieta's Formulas for Cubics and Symmetric Sums

Introduction to Vieta’s Formulas

Vieta’s formulas are one of algebra’s hidden superpowers. They let you explore relationships between the roots of a polynomial and its coefficients without ever having to calculate what those roots actually are. This is incredibly powerful for tackling complex problems that would be nearly impossible using the quadratic formula alone.

The key insight? The coefficients of a polynomial encode information about its roots. Vieta’s formulas translate that encoding for you.

Vieta’s Formulas for Quadratics

Let’s start with what you might already know, then extend it.

Vieta's Formula for Quadratics

Sum of roots:
Product of roots:

Why This Works

When you factor a polynomial with roots and , you get:

Expanding the right side:

Comparing coefficients with :

Coefficient of : , so
Constant term: , so

Warm-up: Quadratic Roots

Problem: Let and be the roots of . Find: (a) (b) (c)

Solution:

Given , we have , , .

(a)

(b)

(c)

Going Deeper: Symmetric Sums and Polynomial Expressions

The real power of Vieta’s formulas emerges when you need to find expressions involving the roots that aren’t immediately obvious. You’ll use a key technique: expressing complicated symmetric functions in terms of the basic symmetric sums.

Question

What’s a symmetric sum?

Answer

An expression in multiple variables that remains unchanged when you swap any two variables. Example:

is symmetric in

and

(swapping them gives the same expression), while

is not.

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Technique: Building Blocks

When you encounter a symmetric expression you need to evaluate, build it from these building blocks:

Elementary symmetric sums: For roots : the sum and product (you get these directly from Vieta’s)
Power sums: , , etc. (derive using identities)
Reciprocal sums: , etc. (rewrite in terms of known quantities)

Finding

Problem: For the quadratic with roots and , find .

Solution:

From :

Using :

More Symmetric Sum Patterns

Working with Reciprocals

Problem: For with roots and , find: (a) (b)

Solution:

From above: ,

(a)

(b) (using from above)

Power Sums and Recurrence Relations

For higher power sums like or , there’s an elegant pattern. If and are roots of , then:

satisfies the recurrence:

This means you can compute any power sum once you have and .

Computing Higher Power Sums

Problem: For with roots and , find: (a) (b) (c)

Solution:

From : ,

(a)

(b)

Vieta’s Formulas for Cubic Polynomials

Now let’s extend to cubic polynomials. The structure is richer, but the principle is exactly the same.

Vieta's Formula for Cubics

Sum of roots:
Sum of products of pairs:
Product of all roots:

Why Cubic Vieta’s Works

Expanding :

Matching coefficients with directly gives the formulas above.

Question

For cubic

, what is

Answer

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Question

For cubic

, what is

Answer

Click to flip • Press Space or Enter

Question

For cubic

, what is

Answer

Click to flip • Press Space or Enter

Applications with Cubics

Direct Application

Problem: Let , , and be the roots of . Find: (a) (b) (c) (d) (e) (f)

Solution:

For : , , ,

(a)

(b)

(c)

(d)

(e) Expanding:

(f)

Advanced Techniques for Cubics

As problems get harder, you’ll need techniques for computing , , and other symmetric functions of three roots.

Computing

Remember the identity for three variables:

Rearranging:

This is the cubic equivalent of what you used for quadratics!

Sum of Squares for Cubics

Problem: For with roots , , , find .

Solution:

From :

Using the identity:

Computing

This is trickier. The key identity is:

Which gives:

Sum of Cubes

Problem: For with roots , , , find .

Solution:

Using , , :

Complex Symmetric Expressions

Once you master the basic techniques, you can tackle intricate combinations. The strategy is always the same: decompose the expression into pieces you know how to evaluate.

Complex Product Expression

Problem: For with roots , , , find .

Solution:

Since each root satisfies , we can view this as finding the product of for all roots .

Note that , so:

Using Vieta’s for : , ,

For , consider that if , then:

For , evaluate:

Therefore:

Key Takeaways

Mastering Vieta's Formulas

Quadratics: Know that and . Build complicated expressions from these.
Cubics: Know all three symmetric sums—they’re your building blocks for everything else.
The Strategy: Identify the elementary symmetric sums from the polynomial, then express what you’re looking for in terms of these.
Powerful Identities: , ,
No roots needed: The entire point is that you never have to compute the actual roots. The polynomial’s coefficients tell you everything you need.

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