iGCSE (2021 Edition)

# 5.02 Expansion

Lesson

Expansion takes an expression from factored form and uses the distributive property to remove the brackets and write the expression as a sum of products. Such as $\left(a+2\right)\left(b+3\right)=ab+3a+2b+6$(a+2)(b+3)=ab+3a+2b+6.

The distributive property: $A\left(B+C\right)=AB+AC$A(B+C)=AB+AC

This can be used over more terms:

$A\left(B+C+D\right)=AB+AC+AD$A(B+C+D)=AB+AC+AD

Or applied multiple times:

 $\left(A+B\right)\left(C+D\right)$(A+B)(C+D) $=$= $A\left(C+D\right)+B\left(C+D\right)$A(C+D)+B(C+D) $=$= $AC+AD+BC+BD$AC+AD+BC+BD

You can reduce this working to one line by remembering the first term multiplies each in the second bracket, then the second term multiplies each in the second bracket.

Hence, $\left(x+5\right)\left(x+2\right)=x^2+2x+5x+10$(x+5)(x+2)=x2+2x+5x+10. Which we could then simplify to $x^2+7x+10$x2+7x+10.

Remember

After expanding check if the expression can be simplified by collecting like-terms.

#### Practice questions

##### Question 1

Expand and simplify the following:

$\left(2n+5\right)\left(5n+2\right)-4$(2n+5)(5n+2)4

##### Question 2

Expand and simplify the following expression $\left(x+7\right)\left(x-7\right)-\left(x-3\right)^2$(x+7)(x7)(x3)2

## Further expansion

When multiplying more than two expressions reduce the problem to multiply two at a time.

#### Practice questions

##### Question 3

Expand the following:

$\left(2x+1\right)\left(5x+7\right)\left(2x-1\right)$(2x+1)(5x+7)(2x1)

##### Question 4

Expand $\left(3c+2\right)\left(2c^2+2\right)^2$(3c+2)(2c2+2)2.