Thirty-Six

Unit 4 · Expressions, Quadratics & PolynomialsA1 · Expand & Factor · Structures

Factoring & Algebraic Identities

Factor common factors, trinomials, and special products quickly and accurately.

  • 30 min
  • Unlocks: Modular Tank Segments
  • Desmos way

Ready when you are

7 steps · about 30 minutes

Start with the warm-up.

Start

Step 1 of 7Warm-upDone

Warm-up

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Key terms

factor
an expression multiplied by another expression

Step 2 of 7Try it firstDone

Try it first

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Try this before the explanation. Commit to a setup, even if you are unsure.

Step 3 of 7LearnDone

The big idea

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The decision that matters

Pull out a greatest common factor before looking for a trinomial pattern. Write that decision before calculating. It turns a crowded ACT stem into a familiar structure and makes the likely distractors easier to spot.

Key idea

Structure before arithmetic

Pull out a greatest common factor before looking for a trinomial pattern. Keep exact values through the setup; round only when the stem asks.
Core relationship
a2−b2=(a−b)(a+b)a^2-b^2=(a-b)(a+b)

Use the formula only after matching every symbol to the diagram or equation.

Algebra or Desmos?

Factor by algebra so the factors remain exact. Graphing the polynomial and proposed factored form provides a quick check: both should share every point and show the same zeros.

Explore in Desmos

Check the factors of 6x2+x−26x^2+x-2

Graph both forms as a quick check; the algebraic grouping proves the factorization.

The calculator is live: change anything and see what happens.

Check the factors of 6x^2+x-2

Degree mode

Press Escape to leave the calculator (twice if a menu is open).

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Trap

The main trap

A zero rr creates factor x−rx-r, so the signs oppose.

Step 4 of 7Worked examplesDone

Worked examples

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Skip check · 2 questions

Already know this? Get both right on the first try and you can skip the worked examples.

Worked example

Example 1

Problem

Factor 6x2−15x6x^2-15x completely.
The greatest common factor is 3x3x.

After dividing each term by 3x3x, what remains inside the parentheses?

6x2−15x=3x(2x−5)6x^2-15x=3x(2x-5).

Which check confirms the factorization?

Worked example

Example 2: ACT twist

Problem

Factor 6x2+x−26x^2+x-2 completely.
Multiply 6(−2)=−126(-2)=-12; use 44 and −3-3.

Which pair correctly splits the middle term while preserving a sum of xx?

Group to get (3x+2)(2x−1)(3x+2)(2x-1).

Why is the split 4x−3x4x-3x valid?

Faded example · you finish it

Example 3: You finish it

Problem

Factor x2−49x^2-49.
x2−72x^2-7^2

Complete the calculation.

Desmos way

Graph the polynomial and its factors

Graph both forms as a quick check; the algebraic grouping proves the factorization.

Graph the polynomial and its factors

Degree mode

Press Escape to leave the calculator (twice if a menu is open).

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Step 5 of 7PracticeDone

Practice

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Step 6 of 7Spot the mistakeDone

Spot the mistake

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Spot the mistake

Where did it go wrong?

One line has an error. Click the line where it first goes wrong.

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Exit ticket

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Optional · 4 questions · Hardest levelChallenge set: the kind of question that decides a 36

These are last-ten-questions hard. Take your time, use Desmos when it helps, and expect to miss some: that's where the learning is.