Mathematics · Expressions, equations and inequalities · Grade 7

Special products (algebraic identities)

A carpet has been widened by b on every side — what are the pieces of (a + b)²? Work out 55² in your head

The situation

The side of a square carpet is a metres. We widen it by b metres on two sides — now the side is a + b. The new area is (a + b)² — but it is not a² + b²! If we draw it, four pieces appear: the old carpet a², two identical rectangles ab and a small square b² in the corner. What if we cut a b × b square from the corner of a square stone slab — can the remaining part be turned into a rectangle? These pictures are the special products themselves: they also help to work out 55² or 101 · 99 in your head.

Topic

Special products: the square of a sum and of a difference, the difference of two squares, cubes, factorising. Extending the border of a square carpet: the pieces of the square (a + b)², a² − b² turns into a rectangle, the cube (a + b)³

Key ideas

(a + b)²
a² − b²
(a + b)³

Formulas

{(a \pm b)}^{2} = a^{2} \pm 2 a b + b^{2}

Square of a sum and of a difference: both squares and twice the product — four pieces in the drawing.

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

Difference of squares: a square with a corner cut off turns into a rectangle.

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

Cube of a sum: the cube splits into 8 pieces — 1 + 3 + 3 + 1.

a^{3} \pm b^{3} = (a \pm b) (a^{2} \mp a b + b^{2})

Sum and difference of cubes: when multiplying, the middle terms cancel in pairs.

Theory

Square of a sum and of a difference

A square of side $a + b$ is divided by two cuts into four pieces: $a^{2}$ , two of $a b$ and $b^{2}$ . So ${(a + b)}^{2} = a^{2} + 2 a b + b^{2}$ — do not forget the double product: ${(a + b)}^{2} \neq a^{2} + b^{2}$ . For the square of a difference: $a b$ is subtracted twice from $a^{2}$ , but the $b^{2}$ in the corner has been subtracted twice — we put it back once: ${(a - b)}^{2} = a^{2} - 2 a b + b^{2}$ . Mental arithmetic: $55^{2} = {(50 + 5)}^{2} = 2500 + 500 + 25 = 3025$ .

Difference of two squares

From a square of side $a$ we cut a $b \times b$ square from the corner: the area is $a^{2} - b^{2}$ . If we split the remaining shape in two with one cut and rotate one piece, we get a rectangle with sides $a - b$ and $a + b$ : $a^{2} - b^{2} = (a - b) (a + b)$ . Mentally: $101 \cdot 99 = (100 + 1) (100 - 1) = 10 000 - 1 = 9 999$ .

Cube of a sum and cube of a difference

A cube of edge $a + b$ is split by three planes into 8 pieces: one $a^{3}$ , three $a^{2} b$ , three $a b^{2}$ and one $b^{3}$ : ${(a + b)}^{3} = a^{3} + 3 a^{2} b + 3 a b^{2} + b^{3}$ . If we put $- b$ in place of $b$ , the terms with an odd power of $b$ change sign: ${(a - b)}^{3} = a^{3} - 3 a^{2} b + 3 a b^{2} - b^{3}$ . The coefficients $1, 3, 3, 1$ are a row of Pascal's triangle.

Sum and difference of cubes

If we multiply out $(a + b) (a^{2} - a b + b^{2})$ , six terms appear, and four of them cancel in pairs: $- a^{2} b$ with $+ a^{2} b$ , $+ a b^{2}$ with $- a b^{2}$ . What remains is $a^{3} + b^{3}$ . Similarly $a^{3} - b^{3} = (a - b) (a^{2} + a b + b^{2})$ . The expression in the second bracket is called an incomplete square ( $a^{2} - a b + b^{2}$ — of a difference, $a^{2} + a b + b^{2}$ — of a sum): it has the plain product, not the doubled one.

Factorising and applications

The formulas factorise in reverse: $x^{2} - 9 = (x - 3) (x + 3)$ ; $4 x^{2} + 12 x + 9 = {(2 x)}^{2} + 2 \cdot 2 x \cdot 3 + 3^{2} = {(2 x + 3)}^{2}$ . To check that a trinomial is a perfect square: are the outer terms squares, and is the middle one twice the product of their roots? Completing the square: $x^{2} + 6 x + 5 = (x^{2} + 6 x + 9) - 4 = {(x + 3)}^{2} - 4$ — a bridge to solving quadratic equations.

About the model

The drawings are made for $a > b > 0$ — only then does every piece have a positive area. The formulas themselves are true for any numbers, negative ones too: they can be proved algebraically by the rule for multiplying polynomials. In the cube model the edges are 1–3 units; the sliders are limited, the formulas are not.

Did you know?

Euclid proved the formula ${(a + b)}^{2} = a^{2} + 2 a b + b^{2}$ geometrically — through the pieces of a square — in Book II of the “Elements” (about 300 BC), Proposition 4.
The triangle of numbers giving the coefficients of ${(a + b)}^{n}$ (1; 1 1; 1 2 1; 1 3 3 1 …) is known in Europe as Pascal's triangle, but it was known several centuries earlier to Omar Khayyam (11th century) and the Chinese mathematician Yang Hui (13th century).

Inside the experiment

The topic is explained step by step with animation, from simple to advanced: from a real-life picture to the formula and the proof. Then you drag the points yourself to check. Inside are missions, a quiz and a guided tour.

More experiments in this section

Expressions: from numbers to letters — Grade 7
Monomials and polynomials — Grade 7
Algebraic fractions — Grade 7
The linear equation: the balance — Grade 7
Systems of equations — Grade 7
Inequalities: ray and interval — Grade 8
The quadratic equation: the parabola — Grade 8