2.3.1 Expanding brackets
Expanding:
Brackets are used to group algebraic equations. The process of removing brackets is known as expanding.
Expanding single bracket
To expand the bracket
Multiply every term inside the bracket by the term on the outside
Collect the like terms
Simplify
Be careful with negative terms
Worked example:
Expand \( -5x(-5+3x) \)
Multiply every term inside the bracket by the term on the outside
\( (-5x) \times (-5) +(-5x) \times (3x) \)
Simplify
\( =25x+(-15x^2) \)
\( =25x-15x^2 \)
\( 25x-15x^2 \)
Expanding multiple brackets
When there is more than one bracket
Expand each bracket separately by multiplying every term inside the bracket by the term on the outside
Collect like terms together
Simplify
Worked example:
Expand \( 2(3x+8)-5(6-4x) \)
Expand both brackets separately by multiplying every term inside the bracket by the term on the outside
\( 2 \times 3x+2 \times 8-5 \times 6-5 \times (-4x) \)
Simplify
\( 6x+16-30+20x \)
Collect like terms
\( 6x+20x+16-30 \)
\( 26x-14 \)
\( 26x-14 \)
Expanding double brackets
To expand the double brackets
Multiply each term in the first bracket by each term in the second bracket
The terms should always be multiplied in the correct order to prevent forgetting any.
First, Outside, Inside and Last (FOIL) is a popular technique.
Use brackets for negative terms
Simplify
Collect like terms
Worked example:
Expand and simplify \( (x+7)(x-3) \).
Using FOIL, multiply both brackets
\( x \times x +x \times (-3)+7 \times x+7 \times (-3) \)
Simplify
\( x^2-3x+7x-21 \)
Collect like terms
\( x^2+4x-21 \)
\( x^2+4x-21 \)
Expanding a squared bracket
For expanding a squared bracket, we may use square identities
\( (a+b)^2=a^2+2ab+b^2 \)
\( (a-b)^2=a^2-2ab+b^2 \)
Alternative method:
Rewrite the squared bracket as two separate brackets multiplied together
e.g.
\( (3x+5y)^2=(3x+5y)(3x+5y) \)
Solve by using FOIL
Simplify
Collect like terms
Worked example:
Expand \( (2x+5)^2 \)
By using square identity \( (a+b)^2=a^2+2ab+b^2 \)
\( (2x+5)^2=2x^2+2(2x)(5)+5^2 ) \)
Simplify
\( 4x^2+20x+25 \)
\( 4x^2+20x+25 \)
Expanding three brackets
To expand three brackets
Multiply any two brackets using an appropriate method
Simplify the result by collecting like terms
Put this result in one bracket
Now multiply it with the third bracket by using an appropriate method
Collect like terms
Simplify
Worked example:
Expand and simplify \( (x-5)(2x+4)(4x-5) \).
Multiply first two brackets using FOIL method
\( (x-5)(2x+4)=(x \times 2x)+(x \times 4)+(-5) \times 2x+(-5) \times 4 \)
Simplify
\( 2x^2+4x-10x-20 \)
Collect like terms
\( 2x^2-6x-20 \)
Put this result in a bracket and multiply it by \( (4x-5) \) using FOIL method
\( (2x^2-6x-20)(4x-5) \)
\( =(2x^2 \times 4x+2x^2 \times (-5)+(-6x) \times 4x+(-6x) \times (-5)+(-20) \times 4x+(-20) \times (-5) ) \)
Simplify
\( 8x^3-10x^2-24x^2+30x-80x+100 \)
Collect like terms
\( 8x^3-34x^2-50x+100 \)