# 1. Special Products

The following special products come from multiplying out the brackets. You'll need these often, so it's worth knowing them well.

a(x+y) =ax+ay(

x+y)(x−y) =x^{2}−y^{2}(Difference of 2 squares)(

x+y)^{2}=x^{2}+ 2xy+y^{2}(Square of a sum)(

x−y)^{2}=x^{2}− 2xy+y^{2}(Square of a difference)

### Examples using the special products

**Example 1:** Multiply out 2*x*(*a* − 3)

**Example 2:** Multiply `(7s + 2t)(7s − 2t)`

**Example 3:** Multiply (12* *+ 5*ab*)(12 − 5*ab*)

**Example 4:** Expand (5*a* + 2*b*)^{2}

**Example 5: **Expand (*q* − 6)^{2}

**Example 6:** Expand (8*x* −* y*)(3*x* + 4*y*)

**Example 7:** Expand (*x* + 2 + 3*y*)^{2}

## Special Products involving Cubes

The following products are just the result of multiplying out the brackets.

(

x+y)^{3}=x^{3}+ 3x^{2}y+ 3xy^{2}+y^{3}(Cube of a sum)(

x−y)^{3}=x^{3}− 3x^{2}y+ 3xy^{2 }−y^{3}(Cube of a difference)(

x+y)(x^{2}−xy+y^{2}) =x^{3}+y^{3 }(Sum of 2 cubes)(

x−y)(x^{2}+xy+y^{2}) =x^{3 }−y^{3 }(Difference of 2 cubes)

These are also worth knowing well enough so you recognize the form, and the differences between each of them. (Why? Because it's easier than multiplying out the brackets and it helps us solve more complex algebra problems later.)

**Example: **Expand `(2s + 3)^3`

### Exercises

Expand:

(1) (*s *+ 2*t*)(*s *− 2*t*)

(2) (*i*_{1} + 3)^{2}

(3) (3*x *+ 10*y*)^{2}

(4) (3*p *− 4*q*)^{2}

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