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Expanding and Factorising

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Expanding Can’tcannot What is ‘expanding’?

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Expanding Yes, that is expanding But what we want to know is how to expand an algebraic equation 3(7 + 10) 5x(6y – 7z) 4(x – 4) + 5

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Expanding Expansion means to multiply everything inside the brackets by what is directly outside the brackets Think Write Write the expression Expand the brackets Multiply out the brackets 4(x – 4) + 5 = 4(x) + 4(-4) + 5 = 4x

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Expanding single brackets After expanding brackets, simplify by collecting any like terms Think Collect any like terms = 4x = 4x - 11 Write

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Your turn Remember: Stop and think Ask your neighbour quietly Hand-up Move on until I can get to you Got a question

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Expanding two brackets Expand each bracket: working from left to right

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Expanding pairs of brackets When multiplying expressions within pairs of brackets, multiply each term in the first bracket by each term in the second bracket, then collect the like terms

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Expanding pairs of brackets You can use the ‘FOIL’ method to help you keep track of which terms are to be multiplied together First – multiply the first term in each bracket Outer – multiply the 2 outer terms Inner – multiply the 2 inner terms Last – multiply the last term of each bracket

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Expansion patterns Difference of two squares (a + b) (a – b) = a 2 – b 2

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Expansion patterns Perfect squares (identical brackets) Square the first term, add the square of the last term, then add (or subtract) twice their product (a + b) (a + b) = a 2 + 2ab + b 2 (a – b) (a – b) = a 2 – 2ab – b 2

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Expanding more than two brackets Brackets or pairs of brackets that are added or subtracted must be expanded separately Always collect any like terms following an expansion

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Factorising Factorising is the ‘opposite’ of expanding, going from an expanded form to a more compact form Factor pairs of a term are numbers and pronumerals which, when multiplied together, produce the original term

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Highest Common Factor The number itself and 1 are factors of every integer The highest common factor (HCF) of given terms is the largest factor that divides into all terms without a remainder

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Factorising using the highest common factor An expression is factorised by finding the HCF of each term, dividing it into each term and placing the result inside the brackets, with the HCF outside the brackets

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Factorising using the difference of two squares rule To factorise a difference of two squares, a 2 – b 2 we use the rule or formula (a + b) (a – b) = a 2 – b 2 in reverse a 2 – b 2 = (a + b) (a – b)

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Factorising using the Difference-of-two-squares rule Look for the common factor first If there is one, factorise by taking it out Rewrite the expression showing the two squares and identifying the a and b parts of the expression

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Factorising using the Difference-of-two-squares rule Factorise, using the rule a 2 – b 2 = (a + b) (a – b)

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Simplifying algebraic fractions Factorise the numerator and the denominator Cancel factors where appropriate

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Simplifying algebraic fractions If two fractions are multiplied, factorise where possible then cancel any factors, one from the numerator and one from the denominator

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Simplifying algebraic fractions If two fractions are divided, remember to multiply the reciprocal of the second fraction before factorising and cancelling

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