Lesson 2.12

Factoring in Quadratic Form

A degree-4 polynomial like may look intimidating — until you notice it's just a disguised quadratic. Substitute and factor like old times.

Introduction

When you see three terms with exponents that follow a 2:1 ratio (like or ), the polynomial is in quadratic form. A simple substitution turns it into a familiar factoring problem.

Past Knowledge

You know how to factor .

Today's Goal

Recognize quadratic form, substitute, factor, then substitute back to get the final answer.

Future Success

This technique reappears in calculus when analyzing polynomial behavior at critical points.

Key Concepts

Recognizing Quadratic Form

A polynomial is in quadratic form if it matches:

The key is the 2:1 exponent ratio. The highest exponent is double the middle exponent.

Examples of Quadratic Form

let
let
let

The Substitution Process

1
Identify the repeated expression and let u equal it
2
Rewrite the polynomial in terms of u
3
Factor the resulting quadratic
4
Substitute the original expression back for u
5
Factor further if possible

Worked Examples

Example 1: Standard Quadratic Form

Basic

Factor completely.

1

Substitute

2

Factor the Quadratic

3

Substitute Back & Factor Further

Both are differences of squares!

Example 2: With Coefficients

Intermediate

Factor .

1

Substitute

2

Factor (AC Method or Trial)

We need factors of that add to : that's and .

3

Substitute Back

Neither factor is a difference of squares, so this is fully factored over the reals.

Example 3: Cubic Exponents

Advanced

Factor .

1

Substitute

, so we get:

2

Factor & Substitute Back

3

Apply Cube Formulas!

Both factors are themselves sum/difference of cubes:

Common Pitfalls

Stopping Too Early

After substituting back, check if each factor can be factored further (difference of squares, sum/difference of cubes). Always factor completely.

Not Seeing the Pattern

Always check: is the highest exponent exactly double the middle exponent? If yes, you can use substitution. If not, try another method.

Real-Life Applications

In physics, the energy levels of quantum systems are often modeled by degree-4 or degree-6 polynomials. Recognizing quadratic form lets physicists factor these to find the discrete energy values — a technique that bridges algebra and quantum mechanics.

Practice Quiz

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