Thirty-Six

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

Polynomial Operations

Combine, add, subtract, and multiply polynomials without losing a sign.

  • 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

like terms
terms with the same variable part and exponent

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

Distribute first; then combine only identical variable parts. 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

Distribute first; then combine only identical variable parts. Keep exact values through the setup; round only when the stem asks.
Core relationship
a(b+c)=ab+aca(b+c)=ab+ac

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

Algebra or Desmos?

Expand and combine like terms by hand when the expression is short. Desmos can compare the original and simplified forms at several inputs, but the distributive property is what proves the forms are identical.

Explore in Desmos

Check two forms of the same polynomial

Enter both forms, then test several x-values. Matching samples checks your algebra but does not prove an identity.

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

Check two forms of the same polynomial

Degree mode

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

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Trap

The main trap

A minus sign before parentheses changes every sign inside.

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

Simplify 3x2−5x+7−(x2+4x−2)3x^2-5x+7-(x^2+4x-2).
Distribute the subtraction: 3x2−5x+7−x2−4x+23x^2-5x+7-x^2-4x+2.

Which terms may be combined because their variable parts match exactly?

The result is 2x2−9x+92x^2-9x+9.

Why did the constant become 7+27+2?

Worked example

Example 2: ACT twist

Problem

Expand and simplify (2x−3)(x+5)(2x-3)(x+5).
Multiply every pair: 2x2+10x−3x−152x^2+10x-3x-15.

How should the two middle terms be combined?

The product is 2x2+7x−152x^2+7x-15.

Why is the coefficient of xx equal to 7?

Faded example · you finish it

Example 3: You finish it

Problem

Simplify 4y−2(3−y)4y-2(3-y).
4y−6+2y4y-6+2y

Complete the calculation.

Desmos way

Numerical identity check

Enter both forms, then test several x-values. Matching samples checks your algebra but does not prove an identity.

Numerical identity check

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.

Step 7 of 7Exit ticketDone

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.