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Four-part batteries · Cubic Championship

Batteries with four parts

These are the biggest batteries in the game. Four parts to work out, then add together. The rules are exactly the same as before. There is just more of it.

The small ³ means multiply by itself three times

  1. x³ = 2 × 2 × 2
    8
  1. x³ = 3 × 3 × 3
    27

One important difference from squaring: A negative x cubed gives a negative answer.

  1. (−2) × (−2) = 4
    first two multiplications give positive
  2. 4 × (−2) = −8
    so (−2)³ = −8

This is different from squaring. (−2)² = 4. Positive. The ³ keeps the negative. The ² removes it.

Key point

² always gives a positive answer. ³ keeps the sign — negative in, negative out.

A four-part battery broken down

Battery: x³ + 2x² + 3x + 4

Part 1

x³

x times x times x

Part 2

2x²

2 times (x times x)

Part 3

3x

3 times x

Part 4

4

just the number 4

Work out each part. Then add them all together.

x³ + 2x² + 3x + 4, with x = 2

x³ + 2x² + 3x + 4 = 26
Your car: x = 2
  1. Part 1 — 2 × 2 × 2 = 8
    Part 1 = 8
  2. Part 2 — 2 × 2 × 2 = 2 × 4 = 8
    Part 2 = 8
  3. Part 3 — 3 × 2 = 6
    Part 3 = 6
  4. Part 4 — just the number
    Part 4 = 4
  5. Add them up
    8 + 8 + 6 + 4 = 26 ✓

x³ + 2x² + 3x + 4, with x = 3

x³ + 2x² + 3x + 4 = 58
Your car: x = 3
  1. Part 1 — 3 × 3 × 3 = 27
    Part 1 = 27
  2. Part 2 — 2 × 3 × 3 = 2 × 9 = 18
    Part 2 = 18
  3. Part 3 — 3 × 3 = 9
    Part 3 = 9
  4. Part 4 — just the number
    Part 4 = 4
  5. Add them up
    27 + 18 + 9 + 4 = 58 ✓

x³ + 2x² + 3x + 4, with x = −2

x³ + 2x² + 3x + 4 = −2
Your car: x = −2
  1. Part 1 — (−2) × (−2) × (−2) = 4 × (−2) = −8
    Part 1 = −8 ³ keeps the negative
  2. Part 2 — 2 × (−2) × (−2) = 2 × 4 = 8
    Part 2 = 8 ² removes the negative
  3. Part 3 — 3 × (−2) = −6
    Part 3 = −6
  4. Part 4 — just the number
    Part 4 = 4
  5. Add them up
    −8 + 8 + (−6) + 4
    = 0 + (−6) + 4
    = −2 ✓

Three easy mistakes

Mistake 1

Thinking (−2)³ is positive

(−2)³ = −8. It is negative. ³ keeps the sign. Only ² removes it.

(−2)³ = −8. (−2)² = 4. Different signs, different results.

Mistake 2

Mixing up ² and ³

x³ = x × x × x. Three multiplications. x² = x × x. Two multiplications.

With x = 3: x² = 9, x³ = 27. Not the same.

Mistake 3

Losing track across four parts

Four parts is a lot to keep in your head.

Write each one out before you add. One part per line. Then add at the bottom.

Try one yourself

Your car:

Work out each part separately, then add them up

    One more challenge

    Next topic

    Decimal values of x

    What if x is 2.5 or −1.5? The steps are the same. The numbers are trickier.
    ← Three-part batteries
    → Decimal values of x Back to the game