Perfect Cubes

1 MIN READ · PRACTISE AT YOUR OWN PACE · FREE

Idea

A cube of a binomial produces four terms, with coefficients following the pattern $1,3,3,1$ . Recognizing the two mixed terms is essential: two cubes alone use a different identity.

Visual guide

VISUAL GUIDEA perfect cube shifts the zero

x³
(x + 1)³

The graph of (x + 1)³ = x³ + 3x² + 3x + 1 is the graph of x³ shifted one unit left. The flattened crossing at x = −1 represents a root repeated three times.

Method

(A+B)^3=A^3+3A^2B+3AB^2+B^3.

(A-B)^3=A^3-3A^2B+3AB^2-B^3.

For subtraction, the signs alternate because odd powers of $-B$ are negative.

Worked example

For $8x^3-36x^2+54x-27$ , try $A=2x$ , $B=3$ . The mixed terms are $-3(2x)^2(3)=-36x^2$ and $3(2x)(3^2)=54x$ . Thus the expression is $(2x-3)^3$ .

PAUSE & THINKA quick check, not a grade

Try it yourself.

Choose an answer and explain your reasoning to yourself. Use a hint if you get stuck.

What is the coefficient of x in

(x+2)^3

Hint 1 · Find a starting point

The expansion has four terms.

Hint 2 · Take the next step

Use x³+3x²a+3xa²+a³ with a = 2.

Show the reasoning

Answer: 12

The x term is 3 × x × 2² = 12x.

Before moving on: what would make one of the other answers wrong? Saying why is part of understanding.

Common mistake

A repeated cubic factor crosses the axis at its zero; it does not behave like a repeated square. Multiplicity matters.

Check your understanding

Factor $t^3+6t^2+12t+8$ .

Show answer

$(t+2)^3$ .

MAKE IT YOURS

Pause before the next idea.

Can you explain how the visualization connects to this lesson’s goal? If a step still feels uncertain, put this lesson on your review list and try the check again another day.

Optional marks, not a grade. Saved in this browser only. Open notebook →