What you need to know

Things to remember:

  • You can use the term “FOIL” to remember the order we multiply.
  • To “expand and simplify” means to multiply out the brackets and then collect like terms.

Before we start, we just need to recall two bits of terminology for multiplying brackets:

  • Expand means multiply out brackets.
  • Simplify means to collect like terms.

 

To start, we will recap how we expand a single bracket, which we can do in one step.

Expand 2x(4x-3).

 

Step 1: Multiply what is inside the bracket by what is outside the bracket.

 

 

So, what was the point, how does this help expand two brackets? Let’s take a look at a very algebraic example.

Expand (a+b)(c+d)

 

Step 1: Multiply what is inside the bracket by what is outside the bracket.

If we look at (c+d) as our “bracket”, then what is outside it? Another bracket! So, we are multiplying everything inside the second bracket by our first bracket.

 

 

But, this doesn’t quite look right, we like to have the letters on the left of the bracket.

 

 

Now, this looks a lot more like what we are used to, expanding single brackets!

 

 

But, we like to have our letters in alphabetical order…

 

 

And finally, if we rearrange it and make it equal to our original double brackets, we can see something pretty neat.

 

 

So, when we multiply double brackets, we multiply the letters in the first bracket by both letters in the first bracket. This method is often referred to as FOIL because you multiply the FIRST letters, then the OUTER letters, then the INNER letters, and finally the LAST letters. Now we can try one with some numbers.

Expand and simplify (x+3)(x-4).

Step 1: Expand the brackets (Do FOIL).

(x+3)(x-4)=x\times x-xtimes4+3\times x -3\times4

(x+3)(x-4)=x^2-4x+3x -12

Step 2: Simplify (Collect the like terms)

(x+3)(x-4)=x^2-x -12

So, not too tricky, but let’s try a trickier one.

Expand and simplify (3x-1)(2x-5).

Step 1: Expand the brackets (Do FOIL).

Hint: Remember, a negative multiplied by a negative is positive, and a negative multiplied by a positive is negative.

(3x-1)(2x-5)=3x\times2x -3xtimes5-1\times2 x +1\times5

(3x-1)(2x-5)=6x^2-15x-2x +5

Step 2: Simplify (Collect the like terms)

(3x-1)(2x-5)=6x^2-17x +5

Example Questions

Question 1: Expand and simplify (x+5)(x+4).

Answer

Step 1: Expand the brackets (Do FOIL).

 

Hint: Remember, a negative multiplied by a negative is positive, and a negative multiplied by a positive is negative.

 

(x+5)(x+4)=x\times x + x\times 4+5\times x +5\times4

 

(x+5)(x+4)=x^2+4x+5x +20

 

Step 2: Simplify (Collect the like terms)

 

(x+5)(x+4)=x^2+9x +20

Question 2: Expand and simplify (7x-4)(3x+1).

Answer

Step 1: Expand the brackets (Do FOIL).

 

Hint: Remember, a negative multiplied by a negative is positive, and a negative multiplied by a positive is negative.

 

(7x-4)(3x+1)=7x\times3x + 7x\times1-4\times 3x -4\times1

 

(7x-4)(3x+1)=21x^2+7x-12x -4

 

Step 2: Simplify (Collect the like terms)

 

(7x-4)(3x+1)=21x^2-5x -4