Cut a rectangle in two
A rectangle 3 squares tall and 5 squares wide holds 3 × 5 = 15 squares. Now cut its width into two parts, 3 and 2. The rectangle splits into two smaller rectangles: one 3 by 3, holding 9 squares, and one 3 by 2, holding 6 squares. Together they still hold 9 + 6 = 15 squares.
Written with a bracket, the width is 3 + 2, so the area is 3 × (3 + 2). Split into its two parts, the same area is 3 × 3 + 3 × 2. So 3 × (3 + 2) = 3 × 3 + 3 × 2. The 3 outside the bracket multiplies each number inside it.
3 × 5 = 3 × 1 + 3 × 4 = 15: the cut moves squares from one part to the other, so the total stays
Cut the 5 columns after 3
Drag the cut along the rectangle. Squares move from one part to the other, but the total stays 3 × 5 = 15. Cut the 5 columns after 3.
The sides can be letters
The cut works whatever the lengths are, so the sides can be letters. A rectangle a tall and x + b wide has area a(x + b). Cut it where x ends and b begins, and it splits into a strip a by x, of area ax, and a strip a by b, of area ab. So a(x + b) = ax + ab.
Rewriting a(x + b) as ax + ab is called expanding the bracket. The rule is that the term outside the bracket multiplies every term inside it. This is the distributive law.
The rectangle a by (x + b) is the ax strip and the ab strip together.
Expanding with numbers
Expand 4(2x + 3). The 4 multiplies both terms inside: 4 × 2x = 8x and 4 × 3 = 12. So 4(2x + 3) = 8x + 12.
It is the same fact you use to multiply 4 × 23 in your head. 23 = 20 + 3, so 4 × 23 = 4 × 20 + 4 × 3 = 80 + 12 = 92.
A rectangle 4 by (2x + 3) is a strip of 8x and a strip of 12.
A letter outside the bracket
The term outside can be a letter too. Expand x(x + 5): and x × 5 = 5x, so .
Minus signs
A minus sign inside the bracket stays with its term. 3(x − 4) = 3 × x − 3 × 4 = 3x − 12.
When the number outside is negative, it multiplies every term with its sign. Expand −2(x − 5): −2 × x = −2x, and −2 × (−5) = +10, because a negative number times a negative number is positive. So −2(x − 5) = −2x + 10.
The usual mistakes
Multiplying only the first term. 5(x + 3) is not 5x + 3. The 5 multiplies the 3 as well, so 5(x + 3) = 5x + 15.
Losing a sign. −2(x − 5) is not −2x − 10: −2 times −5 is +10.
Worked example: A Vegetable Bed Made Longer, Its Area Written Two Ways
Question A vegetable bed is x m long and 5 m wide. It is made 3 m longer. (a) Write the area of the longer bed in two ways, one with a bracket and one without. (b) A second bed is (2x + 1) m long and 4 m wide. Find the total area of the two beds as a single simplified expression.
1.The longer bed is (x + 3) m long and 5 m wide, so its area is 5(x + 3) m2.
The longer bed is (x + 3) m by 5 m, so its area is 5(x + 3) m2. 2.Split the bed into the old part and the new part. The old part is 5 × x = 5x m2 and the new part is 5 × 3 = 15 m2.
The old part is 5 × x = 5x m2 and the new part is 5 × 3 = 15 m2. 3.(a) The two parts make up the whole bed, so 5(x + 3) = 5x + 15. The 5 multiplies every term inside the bracket.
(a) The two parts make up the whole bed: 5(x + 3) = 5x + 15. 4.The area of the second bed is 4(2x + 1) m2. Expand it in the same way: 4 × 2x + 4 × 1 = 8x + 4.
The second bed is 4(2x + 1) = 4 × 2x + 4 × 1 = 8x + 4 m2. 5.Add the two areas and collect like terms: 5x + 15 + 8x + 4 = 13x + 19.
Add the two areas and collect like terms: 5x + 15 + 8x + 4 = 13x + 19. 6.(b) The total area is (13x + 19) m2. Check with x = 2: the beds are 5 × 5 = 25 m2 and 4 × 5 = 20 m2, which is 45 m2 in all, and 13 × 2 + 19 = 45.
(b) The total area of the two beds is (13x + 19) m2.
Answer: (a) 5(x + 3) = 5x + 15; (b) (13x + 19) m2
Common mistakes
- Expanding 5(x + 3) as 5x + 3. The 5 multiplies both parts of the bed, so the new part is 5 × 3 = 15 m2, not 3 m2.
- Adding 5x + 15 + 8x + 4 to get 32x. Only like terms can be added: the x terms make 13x and the numbers make 19.