Vieta's Formulas - Advanced Problem Solving

Understanding Vieta’s Formulas

Vieta’s formulas reveal a beautiful hidden structure in polynomials: they let you find relationships between roots without actually calculating the roots themselves. This is incredibly powerful because it saves time and often reveals elegant patterns that would be hidden in explicit root calculations.

Imagine you’re solving a puzzle where you don’t need to find every piece individually—instead, you just need to know how they fit together. That’s what Vieta’s formulas do for polynomials.

Why Vieta's Formulas Matter

• Time-saving: Find relationships between roots without computing the roots explicitly
• Pattern recognition: Discover elegant algebraic relationships in polynomial problems
• Universal: Work for any polynomial, regardless of the degree
• Elegant math: Connect coefficients directly to symmetric functions of roots

Quick Intuition

Think of a polynomial as a black box. Vieta’s formulas are the secret manual that tells you about what’s inside without having to open it up completely!

Vieta’s Formulas for Quadratics

For a quadratic with roots and :

Let’s verify these formulas work with concrete examples.

Testing Vieta’s with the Quadratic Formula

Let and be the two roots of .

Problem: (a) Find using the quadratic formula; (b) Find using the quadratic formula; (c) Find using the quadratic formula

Solution:

Using : , ,

So and

(a)

(b)

(c)

Now here’s the magic—we can get the same results instantly using Vieta’s formulas!

Vieta’s Formula Speed Run

Let and be the two roots of . Using Vieta’s formulas, find: (a) ; (b) ; (c) ; (d)

Hints

• For Vieta’s: and (in )
• Parts (a) and (b) are direct applications
• Part (c): Use the identity
• Part (d): Expand

Solution:

For : , ,

(a) ✓ (matches quadratic formula!)

(b) ✓ (matches quadratic formula!)

(c) ✓

(d)

The Power of Vieta's

Notice how Vieta’s formulas give us the answers instantly in parts (a) and (b). No messy quadratic formula calculations needed! And then we can build up to more complex expressions using just algebra.

Proving Vieta’s Formulas

Understanding the proof helps you remember why these formulas work.

Prove Vieta’s Formula for Quadratics

Problem: Let , , and be real numbers with . If the quadratic has roots and , prove that:

Hints

• Key fact: If a polynomial has roots and , you can write it as
• Expand and compare to
• Match coefficients of like powers

Solution:

Since and are roots of , we can factor:

Expand the right side:

Now compare coefficients with :

Coefficient of : ✓

Coefficient of : , so ✓

Constant term: , so ✓

Building Intuition with Quadratic Examples

Now let’s apply Vieta’s formulas to more complex expressions involving roots.

Quadratic Roots Practice

Let and be the two roots of . Find: (a) ; (b) ; (c) ; (d)

Hints

• From Vieta’s: and
• Part (a): Use
• Part (b): Expand and use
• Part (c): Find common denominator:
• Part (d): Use

Solution:

From Vieta’s formulas: and

(a)

(b)

(c)

(d)

Power Sums: Finding Higher Powers

One of the most elegant applications of Vieta’s formulas is computing power sums like without finding and explicitly.

Power Sums for Quadratics

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

Hints

• From Vieta’s: and
• For powers: Use recursion! If , then
• For (b): Use
• For reciprocals: Find the pattern using Vieta’s on the reciprocals

Solution:

From Vieta’s: and

(a)

(b)

(c) Using the recursion :

(d)

(e)

(f)

Power Sum Trick

• Recursion formula:
• Cubes:
• Reciprocals:

Extending to Cubic Polynomials

The beauty of Vieta’s formulas is that they generalize to any polynomial degree!

Prove Vieta’s Formula for Cubics

Problem: Let , , , and be real numbers with . If the cubic has roots , , and , prove that:

Hints

• Factor form:
• Expand step by step: first , then multiply by
• Match coefficients with the original polynomial

Solution:

Since , , are roots:

Expand

Then multiply by :

So:

Comparing with :

• Coefficient of : → ✓
• Coefficient of : → ✓
• Constant: → ✓

Cubic Applications: Building Complex Expressions

Now let’s see Vieta’s formulas shine with cubic polynomials and increasingly complex expressions.

Cubic Roots Practice 1

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

Hints

• Apply Vieta’s directly for (a), (b), (c)
• For (d):
• For (e): Expand and recognize the pattern
• For (f): Expand

Solution:

For : coefficients give us

(a)

(b)

(c)

(d)

(e)

(f)

Cubic Roots Practice 2

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

Hints

• From Vieta’s: , ,
• For (a):
• For (b): Use
• For (d): Find common denominator and simplify
• For (f): Expand step by step, using power sums

Solution:

From Vieta’s: , ,

(a)

(b) Using :

(c)

First:

So:

Thus:

(d)

Numerator

So:

(e) Using the identity :

(f)

Where: , and

Advanced Challenge: Triple Root Problem

Here’s a challenging cubic problem that combines multiple techniques.

Cubic Roots Challenge

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

Hints

• From Vieta’s: , ,
• Notice that simplifies many expressions!
• For (d): Use substitution with
• For (e): Think about evaluating the polynomial at and

Solution:

From Vieta’s: , ,

(a)

(b) Since , we have , so .

Using :

(c)

, and

So:

Thus:

(d)

(e) Note that , , are roots of .

At : , so

At : , so

(Note: )

Key Takeaways and Memory Aids

Vieta's Formulas Summary

• For quadratics with roots , : ,
• For cubics with roots , , : Use all three relationships from coefficients
• Power sums: Use recursion to find
• Special cases: When , many expressions simplify dramatically
• Clever substitutions: Evaluating polynomials at specific points reveals products of roots

Question

What is Vieta’s formula for the sum of roots in ?

Answer

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Question

What is Vieta’s formula for the product of roots in ?

Answer

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Question

How do you find if you only know and ?

Answer

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Question

What is the power sum recursion formula for finding ?

Answer

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Learning Checklist

Track your mastery of Vieta’s formulas as you work through these problems.

Vieta's Formulas Mastery Checklist

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Final Reflections

Vieta’s formulas represent one of the most elegant shortcuts in algebra. By connecting the visible parts of a polynomial (its coefficients) to its invisible roots, they let you discover relationships without doing expensive calculations. This is the essence of mathematical maturity—finding the right perspective that makes a problem simple.

As you practice these problems, you’ll develop an intuition for which Vieta identity to apply and when to use algebraic tricks like the power sum recursion. That intuition is worth far more than memorizing formulas.

Good luck, and remember: the best problems are the ones where you find an elegant solution that makes everyone else say, “Why didn’t I think of that?”