Algebra and functions.

Simplifying experession by collecting like terms:

a) 3x+2xy=7-x+3xy-9
= 3x-x+2xy+3xy+7-9
= 2x+5xy-2

b) 3x²-6x+4-2x²+3x-3
=3x²-2x²-6x+3x+4-3
=x²-3x+1

c) 3(a+b²)-2(3a-4b²)
=3a+3b²-6a+8b²
=-3a+11b²

Simplifying experessions and functions by using rules of indices:

a^m x a^n = a^m+n
a^m ÷ a^n = a^m-n
(a^m)^n = a^mn
a^-m = 1/a^m
a^1/m = m√a
a^n/m = m√a^n

a) x²×x⁵
= x²+⁵
= x⁷

b) 2r²x3r³
= 2x3xr²xr³
= 6r⁵

c) b⁴÷b⁴
=b⁴-⁴
=b⁰ = 1

d) 6x^-³÷3x^-⁵
=6÷3×x^-³÷x^-5
=2×x²
=2x²

e) (a³)²x2a²
=a⁶x2a²
=2xa⁶+²
=2a⁸

f) (3x²)³÷x⁴
=27x⁶÷x⁴
=27÷1×x⁶÷x⁴
=27×x⁶-⁴
=27x² 

Expaning expressions by multipyling each term inside the bracket by the term outside:

a) 5(2x+3)
=10x+15

b) -3x(7x-4)
=-21x²+12x

c) y²(3-2y³)
=3y²-2y⁵

d) 4x(3x-2x²+5x³)
=12x²-8x³+20x⁴

e) 2x(5x+3)-5(2x+3)
=10x²+6x-10x-16
=10x²-4x-15 

Factorising expressions:

Factorising is the opposite of expanding expressions.

a) 3x+9
=3(x+3)

b) x² -5x
=x(x-5)

c) 8x² +20x
=4x(2x+5)

d) 9x² y+15xy²
=3xy(3x+5y)

e) 3x² -9xy
=3x(x-3y)

Factorising quadratic expressions:

A quadractic expression has the form ax² +bx+c, where a,b, and c are constants and a doesn't equal 0.
x² -y² =(x+y)(x-y) is the difference of 2 squares.

a) 6x² +9x
=3x(2x+3)

b) x² -5x-6
=(x-6)(x+1)

c) x² +6x+8
=(x+2)(x+4)

d)6x² -11x-10
=(3x+2)(2x-5)

e) x² -25
=x² -5²
=(x+5)(x-5)

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