Four terms, taken two at a time
Factor xy + 3x + 2y + 6. No single factor divides all four terms: x is a factor of xy and 3x, but not of 2y or 6. So there is no single factor to take outside a bracket.
Instead, take the terms two at a time: (xy + 3x) + (2y + 6). Each pair does have a common factor of its own.
Factor each pair
Take x out of the first pair: xy + 3x = x(y + 3). Take 2 out of the second pair: 2y + 6 = 2(y + 3). So the expression is x(y + 3) + 2(y + 3).
Both pairs have left the same bracket behind, y + 3. That is what makes the method work.
Take out the shared bracket
Now there are two terms, x(y + 3) and 2(y + 3), and both contain the factor (y + 3). Take it outside, just as 4 was taken out of 4a + 4b: x lots of (y + 3) and 2 more lots of (y + 3) make (x + 2) lots of (y + 3). So xy + 3x + 2y + 6 = (y + 3)(x + 2).
Check with numbers. When x = 1 and y = 1, xy + 3x + 2y + 6 is 1 + 3 + 2 + 6 = 12, and (y + 3)(x + 2) is 4 × 3 = 12.
The four terms are the four pieces of one rectangle. The colored top row is x(y + 3) and the bottom row is 2(y + 3). Both rows are y + 3 long, so the whole rectangle is (x + 2) by (y + 3).
The same three moves in letters
Factor ax + bx + kay + kby. Pair the terms: (ax + bx) + (kay + kby). Take x out of the first pair and ky out of the second: x(a + b) + ky(a + b). The matching bracket is a + b, so take it out: (a + b)(x + ky).
Minus signs and the order of the terms
Sometimes the second pair needs a negative factor taken out. In xy + 3x − 2y − 6, take x out of the first pair and −2 out of the second: x(y + 3) − 2(y + 3), because −2 × y = −2y and −2 × 3 = −6. The brackets match, so the answer is (y + 3)(x − 2). Taking out +2 instead would give 2(−y − 3), which does not match y + 3.
If the pairs share nothing, change the order of the terms. In xy + 6 + 3x + 2y, the pair xy and 6 has no common factor. Rewrite it as xy + 3x + 2y + 6 first, so that each pair shares a factor.
The usual mistakes
Stopping halfway. x(y + 3) + 2(y + 3) is still a sum of two terms. Take out the shared bracket to finish: (y + 3)(x + 2).
Mixing up the brackets. The 2 taken out of the second pair goes with the x, so the factors are (x + 2) and (y + 3), not (x + 3) and (y + 2). Numbers show the difference: when x = 2 and y = 1, xy + 3x + 2y + 6 is 2 + 6 + 2 + 6 = 16, and (x + 2)(y + 3) is 4 × 4 = 16, but (x + 3)(y + 2) is 5 × 3 = 15.
Worked example: A Drawer Organizer with Four Compartments, and the Sides of the Tray from Its Area
Question A drawer organizer is a rectangular tray divided into four rectangular compartments. Its largest compartment is x cm long and y cm wide, and the maker gives the floor area of the whole tray as (xy + 8x + 6y + 48) cm2. (a) Factor the area by grouping to find the two sides of the tray in terms of x and y. (b) A customer orders a square organizer whose largest compartment is 20 cm long. Find the width of that compartment and the floor area of the tray.
1.The four terms have no common factor, so group them in pairs: (xy + 8x) + (6y + 48).
No factor divides all four terms, so take them in pairs: (xy + 8x) + (6y + 48). 2.Take x out of the first pair and 6 out of the second: x(y + 8) + 6(y + 8). Both pairs leave the same bracket, y + 8.
Take x out of the first pair and 6 out of the second: x(y + 8) + 6(y + 8). 3.(a) Take out the common bracket: xy + 8x + 6y + 48 = (x + 6)(y + 8). The sides of the tray are (x + 6) cm and (y + 8) cm. Check by expanding: (x + 6)(y + 8) = xy + 8x + 6y + 48.
(a) The shared bracket comes out: the sides of the tray are (x + 6) cm and (y + 8) cm. 4.A square tray has two equal sides, so x + 6 = y + 8. The largest compartment is 20 cm long, so x = 20, which gives 26 = y + 8 and y = 18.
A square tray has equal sides, so x + 6 = y + 8, and x = 20 gives y = 18. 5.(b) The largest compartment is 18 cm wide. The tray is 26 cm by 26 cm, so its floor area is 26 × 26 = 676 cm2. Check: 20 × 18 + 8 × 20 + 6 × 18 + 48 = 360 + 160 + 108 + 48 = 676.
(b) The compartment is 18 cm wide, and the floor area of the tray is 26 × 26 = 676 cm2.
Answer: (a) (x + 6) cm and (y + 8) cm; (b) the compartment is 18 cm wide, and the floor area of the tray is 676 cm2
Common mistakes
- Pairing xy with 48 and 8x with 6y. Those pairs share no factor, so no common bracket appears. Pair the terms so that each pair has a factor in common: xy with 8x, and 6y with 48.
- Writing the sides as (x + 8) and (y + 6). Expanding gives xy + 6x + 8y + 48, which is a different area. The 6 taken out of the second pair joins the x, so the sides are (x + 6) and (y + 8).