A4: simplify and manipulate algebraic expressions (including those involving surds and algebraic fractions) by: ● collecting like terms ● multiplying a single term over a bracket ● taking out common factors ● expanding products of two or more binomials ● factorising quadratic expressions of the form x^2 + bx + c, including the difference of two squares; factorising quadratic expressions of the form ax^2 + bx + c ● simplifying expressions involving sums, products and powers, including the laws of indices

Collect like terms: y+x-2z+4xy+6y-z-3x+x²

Collect like terms: y+x-2z+4xy+6y-z-3x+x²

7y-2x-3z+4xy+x²

Multiplying a single term over a bracket: 5(x+3)

5x+15

Multiplying a single term over a bracket: 6x²(x-y-2)

6x³-6x²y-12x²

Taking out common factors: 4x+6c-2x²

2(2x+3c-x²) or 2x(2-x)+6c

Taking out common factors: 3x²-6x-18=0

x²-2x-6=0 (as the equation=0, the left side can be multiplied/divided however)

Expanding products of two or more binomials (expanding two brackets): (2x-1)(x+3)

2x²+6x-x-3

2x²+5x-3

Expanding products of two or more binomials: (x-8)(y-4)

xy-8y-4x+32

Factorising quadratic expressions: x²+9x+20

(X+4)(x+5)

Factorise x²−6x−27

(X+3)(x-9)

Factorise x²-36

(X+6)(x-6) - difference of two squares (x+a)(x-a)

Factorise fully 2y²−50

2y²-50/2=y²-25 (now is difference of two squares)

(Y+5)(y-5)

Factorise 2y²+7y−15

2y²-10y+3y-15

2y(y-5)+3(y-5)

(2y+3)(y-5)

Factorise 7x²−19x+12

7x²-12x-7x+12

7x(x-1)-12(x-1)

(7x-12)(x-1)

Simplifying indices

• A² means axa. A has been multiplied by itself twice. The index, or power, here is 2.

• A³ means axaxa. A has been multiplied by itself three times.

• A^4 means axaxaxa. A has been multiplied by itself four times

What are the laws of indices?

The rules for simplifying calculations or expressions involving powers of the same base. This means that the larger number or letter must be the same across indices to simplify them.

Multiplying indices

A^m+A^n=A^m+n

Dividing indices

A^m-A^n=A^m-n

Simplify 3^5-3³

3²

Indices with brackets

(A^m)^n=A^mxn

Simplify (7^4)²

7^8

Power of 0

Any number to the power of 0 equals 1

Negative indices

A^-m=1/A^m

Simplify 3^-3

1/3³=1/27

Fractional indices

A^m/n= (n√A)^m

Simplify 27^2/3

3√27=3

3²=9