Perfect Cube of a Binomial: Extended Formulas

Article Contents

1. Formula for (a + b)³
2. Why the Formula Works
3. Examples
4. Formula for (a - b)³
5. Common Mistakes
6. Interactive Exercises

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1. Formula for (a + b)³ {#formula-sum}

Cube of a sum of two terms:

$$(a+b{)}^3=a^3+3a^{2}b+3a{b}^2+b^3$$

Pascal's Triangle

Power	Coefficients
$(a+b{)}^0$	$1$
$(a+b{)}^1$	$1,1$
$(a+b{)}^2$	$1,2,1$
$(a+b{)}^3$	$1,3,3,1$

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2. Why the Formula Works {#why-it-works}

$$(a+b{)}^3=(a+b)(a+b)(a+b)$$

Or: $(a+b{)}^3=(a+b)(a+b{)}^2=(a+b)(a^2+2ab+b^2)$

$$=a^3+2a^{2}b+a{b}^2+a^{2}b+2a{b}^2+b^3$$
$$=a^3+3a^{2}b+3a{b}^2+b^3$$

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3. Examples {#examples}

Example 1: Numerical

$$(2+3{)}^3=2^3+3\cdot 2^2\cdot 3+3\cdot 2\cdot 3^2+3^3$$
$$=8+36+54+27=125$$

And indeed: $5^3=125$ ✓

Example 2: Variables

$$(x+2{)}^3=x^3+3x^2(2)+3x(2^2)+2^3$$
$$=x^3+6x^2+12x+8$$

Example 3: Two Variables

$$(a+b{)}^3=a^3+3a^{2}b+3a{b}^2+b^3$$

Example 4: $(x+1{)}^3$

$$(x+1{)}^3=x^3+3x^2+3x+1$$

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4. Formula for (a - b)³ {#formula-difference}

Cube of a difference:

$$(a-b{)}^3=a^3-3a^{2}b+3a{b}^2-b^3$$

Signs alternate: $+-+-$

Example

$$(x-2{)}^3=x^3-3x^2(2)+3x(2^2)-2^3$$
$$=x^3-6x^2+12x-8$$

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5. Common Mistakes {#common-mistakes}

❌ Mistake 1: Wrong Coefficients

$$(a+b{)}^3\ne a^3+b^3$$

The middle terms $3a^{2}b+3a{b}^2$ are missing!

❌ Mistake 2: Sign Confusion

$$(a-b{)}^3=a^3-b^3\text{(WRONG!)}$$
$$(a-b{)}^3=a^3-3a^{2}b+3a{b}^2-b^3\text{(CORRECT)}$$

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6. Interactive Exercises {#interactive-exercises}

Practice cube of binomial:

• Perfect Cube
• Complete the Square
• Expanding

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Formula Summary

Formula	Form
$(a+b{)}^3$	$a^3+3a^{2}b+3a{b}^2+b^3$
$(a-b{)}^3$	$a^3-3a^{2}b+3a{b}^2-b^3$
Sum of cubes	$a^3+b^3=(a+b)(a^2-ab+b^2)$
Difference of cubes	$a^3-b^3=(a-b)(a^2+ab+b^2)$

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Related Articles

• Complete the Square – $(\text{number}{)}^2$
• Difference of Squares – $(\text{number}{)}^2-(\text{other}{)}^2$
• Introduction to Algebra – basics