Algebra - Maths

16 - Ways to simplify algebraic expressions

1. Using index laws -

   • Multiplying the same algebra with indices means you add them.

   • Dividing the same algebra with indices means you subtract them

   • Having the indices as algebra with the larger algebra inside the brackets means you combine them.

2. Squaring or cubing a whole expression.

   • Take each value in the brackets and use the outer index on them. Combine all your answers.

3. Working with negative or fractional indices.

   • A number to the power of anything negative is always equal to 1 over the number to the power of the positive version of the negative index

16 - What to remember when multiplying expressions

1. Multiply any number parts first.

2. Add the powers of each letter to work out the new power.

16 - What to remember when dividing expressions

1. Divide any number parts first

2. Subtract the powers of each letter to work out the new power.

17 - How to expand brackets

Multiply what is outside the bracket by everything inside the bracket.

• When expanding brackets with negative signs outside:

  • Negative signs belong to their terms to the right of them.

17 - Methods for expanding two brackets

1. Using the grid method:

   • The outside of the grid should contain the two terms on either side in the same column / row.

   • Cross multiply between the terms

   • Place the worked out values in the correct order.

2. Use the acronym FOIL:

   • First terms

   • Outer terms

   • Inner terms

   • Last terms

     • For each acronym, multiply them with each other.

17 - When there is a term next to two brackets.

• Put brackets around the whole expansion

• Expand the equation

• Multiply all the terms in the brackets with the term outside the brackets.

18 - Factorising

The opposite of expanding.

18 - What to remember when factorising an expression

You always need to look for the largest factor you can take out from any term.

18 - What to remember when factorising quadratics where there is 𝑥²

• The result should always contain two brackets.

• You need to find two numbers which add to get the second value and multiply to get the third value.

18 - The outcome of b and c when factorising 𝑥² = bx + c

| b | c | Factors |

| Pos | Pos | Both numbers are positive |

| Pos | Neg | Bigger number is positive and smaller number is negative. |

| Neg | Neg | Bigger number is negative and smaller number is positive. |

| Neg | Pos | Both numbers are negative |

18 - What to remember when factorising quadratics where there is a𝑥²

• One of the brackets has to contain the ax term while the other contains x

• Try two numbers which have the product of a × c.

• Check each pair by mutuiplying the brackets.

18 - Difference of two squares

Where the positive and negative versions of a number multiply to get the negative version of the square number. x² should be there for it to work.

Use this rule: a² - b² = (a + b) (a - b)

19 - How to solve a linear equation

You need to get the letter on its own on one side. It is also important to write your work neatly.

1. Every line of working should include an equal sign because equations always have results.

2. Write down the operation you are carrying out. Remember to do the same thing for both sides of the equation.

3. Start a new line for each step. Do one operation at a time

4. Line up the equals signs

19 - How to solve a linear equation when there is a letter on both sides

You need to get the letter on its own on one side of the equation.

1. Start by collecting like terms so that all letters are together.

2. Always get rid of the smaller 𝑥 by adding or subtracting it by the same value to equal zero.

3. Add ot subtract the non 𝑥 to get zero.

4. Divide both sides by the value of 𝑥.

19 - How to answer a linear equation which includes brackets

1. Start by multiplying out the brackets then collecting like terms.

2. Always get rid of the smaller 𝑥 by adding or subtracting it by the same value to equal zero.

3. Add ot subtract the non 𝑥 to get zero.

4. Divide both sides by the value of 𝑥.

19 - What to avoid when solving linear equations

Never use a trial and improvemnt method. You probably can’t find the correct answer and you can’t get any method marks.

47 - How to simplify an algebraic fraction

• Cancel out any like terms

OR

• Divide the numerator and the denominator by what can go into both of them.

47 - What to remember before simplifying algebraic fractions

Factorise the numerator or the denominator if they have more than one term.

47 - How to add or subtract algebraic fractions which have different denominators

1. Find a common denominator.

2. Add or subtract the numerators.

3. Simplify (only if required).

47 - How to multiply algebraic fractions

1. Multiply the numerators and multiply the denominators.

2. Simplify (only if required.)

(Don’t expand the brackets. Its is a lot easier to simplify with them in place.)

47 - How to divide algebraic fractions

1. Change the second fraction to its reciprocal

2. Change the divide sign to the multiply sign

3. Multiply the fractions and simplify.