IntroductionThis chapter focuses on basic manipulation of Algebra

It also goes over rules of Surds and Indices

It is essential that you understand this whole chapter as it links into most of the others!

Algebra and FunctionsLike Terms

You can simplify expressions by collecting like terms

Like Terms are terms that are the same, for example;

5x and 3xb2 and -2b27ab and 8ab

are all like terms.1AExamplesa)b)c)Expand each bracket first

Algebra and FunctionsIndices (Powers)

You need to be able to simplify expressions involving Indices, where appropriate.1B

Algebra and FunctionsIndices (Powers)

You need to be able to simplify expressions involving Indices, where appropriate.1BExamplesa)b)c)d)e)f)

Algebra and FunctionsExpanding Brackets

You can expand an expression by multiplying the terms inside the bracket by the term outside.1CExamplesa)b)c)d)e)

Algebra and FunctionsFactorising

Factorising is the opposite of expanding brackets. An expression is put into brackets by looking for common factors.1Da)Common Factor3b)xc)4xd)3xye)3x

Algebra and FunctionsExpand the following pairs of brackets

(x + 4)(x + 7) x2 + 4x + 7x + 28 x2 + 11x + 28(x + 3)(x 8) x2 + 3x 8x 24 x2 5x - 24+ 28+ 7x+ 7+ 4xx2x+ 4x- 24- 8x- 8+ 3xx2x+ 3x

Algebra and Functionsx2+3x2+You get the last number in a Quadratic Equation by multiplying the 2 numbers in the bracketsYou get the middle number by adding the 2 numbers in the brackets(x + 2)(x + 1)

Algebra and Functionsx2-2x15-You get the last number in a Quadratic Equation by multiplying the 2 numbers in the bracketsYou get the middle number by adding the 2 numbers in the brackets(x - 5)(x + 3)

Algebra and Functionsx2 - 7x + 12Numbers that multiply to give + 12+3 +4-3 -4+12 +1-12 -1+6 +2-6 -2Which pair adds to give -7?(x - 3)(x - 4)So the brackets were originally

Algebra and Functionsx2 + 10x + 16Numbers that multiply to give + 16+1 +16-1 -16+2 +8-2 -8+4 +4-4 -4Which pair adds to give +10?(x + 2)(x + 8)So the brackets were originally

Algebra and Functionsx2 - x - 20Numbers that multiply to give - 20+1 -20-1 +20+2 -10-2 +10+4 -5-4 +5Which pair adds to give - 1?(x + 4)(x - 5)So the brackets were originally

Algebra and FunctionsFactorising Quadratics

A Quadratic Equation has the form;

ax2 + bx + c

Where a, b and c are constants and a 0.

You can also Factorise these equations.

REMEMBER An equation with an x2 in does not necessarily go into 2 brackets. You use 2 brackets when there are NO Common Factors1EExamplesa)The 2 numbers in brackets must: Multiply to give c Add to give b

Algebra and FunctionsFactorising Quadratics

A Quadratic Equation has the form;

ax2 + bx + c

Where a, b and c are constants and a 0.

You can also Factorise these equations.1EExamplesb)The 2 numbers in brackets must: Multiply to give c Add to give b

Algebra and FunctionsFactorising Quadratics

A Quadratic Equation has the form;

ax2 + bx + c

Where a, b and c are constants and a 0.

You can also Factorise these equations.1EExamplesc)The 2 numbers in brackets must: Multiply to give c Add to give b(In this case, b = 0)This is known as the difference of two squares x2 y2 = (x + y)(x y)

Algebra and FunctionsFactorising Quadratics

A Quadratic Equation has the form;

ax2 + bx + c

Where a, b and c are constants and a 0.

You can also Factorise these equations.1EExamplesd)The 2 numbers in brackets must: Multiply to give c Add to give b

Algebra and FunctionsFactorising Quadratics

A Quadratic Equation has the form;

ax2 + bx + c

Where a, b and c are constants and a 0.

You can also Factorise these equations.1EExamplesd)The 2 numbers in brackets must: Multiply to give c Add to give b Sometimes, you need to remove a common factor first

Algebra and FunctionsExpand the following pairs of brackets

(x + 3)(x + 4) x2 + 3x + 4x + 12 x2 + 7x + 12(2x + 3)(x + 4) 2x2 + 3x + 8x + 12 2x2 + 11x + 12+ 12+ 4x+ 4+ 3xx2x+ 3x+ 12+ 8x+ 4+ 3x2x2x+ 32xWhen an x term has a 2 coefficient, the rules are different2 of the terms are doubled So, the numbers in the brackets add to give the x term, WHEN ONE HAS BEEN DOUBLED FIRST

Algebra and Functions2x2 - 5x - 3Numbers that multiply to give - 3-3 +1

+3 -1One of the values to the left will be doubled when the brackets are expanded(2x + 1)(x - 3)So the brackets were originally-6 +1-3 +2+6 -1+3 -2The -3 doubles so it must be on the opposite side to the 2x

Algebra and Functions2x2 + 13x + 11Numbers that multiply to give + 11+11 +1

-11 -1One of the values to the left will be doubled when the brackets are expanded(2x + 11)(x + 1)So the brackets were originally+22 +1+11 +2-22 -1-11 -2The +1 doubles so it must be on the opposite side to the 2x

Algebra and Functions3x2 - 11x - 4Numbers that multiply to give - 4+2 -2

-4 +1

+4 -1One of the values to the left will be tripled when the brackets are expanded(3x + 1)(x - 4)So the brackets were originally+6 -2+2 -6-12 +1-4 +3The -4 triples so it must be on the opposite side to the 3x+12 -1+4 -3

Algebra and FunctionsExtending the rules of IndicesThe rules of indices can also be applied to rational numbers (numbers that can be written as a fraction)1FExamplesa)b)c)d)

Algebra and FunctionsExtending the rules of IndicesThe rules of indices can also be applied to rational numbers (numbers that can be written as a fraction)1FExamplesa)b)c)d)

Algebra and FunctionsExtending the rules of IndicesThe rules of indices can also be applied to rational numbers (numbers that can be written as a fraction)1FExamplesa)b)

Algebra and FunctionsSurd Manipulation

You can use surds to represent exact values.

1GExamplesSimplify the followinga)b)c)

Algebra and FunctionsRationalising

Rationalising is the process where a Surd is moved from the bottom of a fraction, to the top.

1HMultiply top and bottom byMultiply top and bottom byMultiply top and bottom byExamplesRationalise the followinga)

Algebra and FunctionsRationalising

Rationalising is the process where a Surd is moved from the bottom of a fraction, to the top.

1HMultiply top and bottom byMultiply top and bottom byMultiply top and bottom byExamplesRationalise the followingb)

Algebra and FunctionsRationalising

Rationalising is the process where a Surd is moved from the bottom of a fraction, to the top.

1HMultiply top and bottom byMultiply top and bottom byMultiply top and bottom byExamplesRationalise the followingc)

SummaryWe have recapped our knowledge of GCSE level maths

We have looked at Indices, Brackets and Surds

Ensure you master these as they link into the vast majority of A-level topics!