Factorising (Edexcel IGCSE Maths A)

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  • Describe how to factorise simple expressions such as 6 x plus 8.

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Cards in this collection (28)

  • Describe how to factorise simple expressions such as 6 x plus 8.

    To factorise simple expressions like 6 x plus 8:

    1. Find the highest common factor of 6 and 8 (which is 2)

    2. Write this factor outside a set of brackets

    3. Write inside the brackets what you must multiply the factor by to get the original expression

    So 6 x plus 8 becomes 2 open parentheses 3 x plus 4 close parentheses.

  • True or False?

    2 x is the highest common factor in the expression 4 x squared plus 12 x.

    False.

    The highest common factor in 4 x squared plus 12 x is 4 x, not 2 x.

  • True or False?

    Factorisation can be thought of as the opposite of expanding brackets.

    True.

    Factorisation can be thought of as the opposite of expanding brackets.

  • An expression is factorised to get 5 open parentheses x minus 1 close parentheses.

    How can this result be checked?

    This result can be checked by expanding the expression.

    So 5 open parentheses x minus 1 close parentheses expands to give 5 x minus 5.

    If that was the original question, then the factorised expression is correct.

  • True or False?

    You can factorise out negative numbers.

    E.g. negative 2 can be factorised out from the expression negative 2 x minus 4.

    True.

    You can factorise out negative numbers.
    Just be very careful with the signs.

    E.g. negative 2 x minus 4 factorises to negative 2 open parentheses x plus 2 close parentheses.

  • True or False?

    The expression 3 x open parentheses 5 x plus 10 close parentheses is factorised fully.

    False.

    The expression 3 x open parentheses 5 x plus 10 close parentheses is not factorised fully.

    You can still take out a 5 from inside the brackets.

    This gives 15 x open parentheses x plus 2 close parentheses, which is now factorised fully.

  • True or False?

    It is possible to factorise open parentheses x minus 3 close parentheses out of the expression 2 open parentheses x minus 3 close parentheses plus y open parentheses x minus 3 close parentheses.

    True.

    It is possible to factorise open parentheses x minus 3 close parentheses out of the expression 2 open parentheses x minus 3 close parentheses plus y open parentheses x minus 3 close parentheses. You can treat the open parentheses x minus 3 close parentheses as if it were a single term.

    This gives open parentheses x minus 3 close parentheses open parentheses 2 plus y close parentheses.

  • Write down the first step when factorising the expression x y plus 3 y plus 2 x plus 6.

    The first step when factorising x y plus 3 y plus 2 x plus 6 is to fully factorise the first pair of terms and fully factorise the last pair of terms.

    This gives y open parentheses x plus 3 close parentheses plus 2 open parentheses x plus 3 close parentheses.

  • True or False?

    You get the same result when factorising x y plus 3 y plus 2 x plus 6 as you do when swapping the middle terms and factorising x y plus 2 x plus 3 y plus 6.

    True.

    You get the same result when factorising x y plus 3 y plus 2 x plus 6 as you do when swapping the middle terms and factorising x y plus 2 x plus 3 y plus 6.

    For x y plus 3 y plus 2 x plus 6, you can factorise out open parentheses x plus 3 close parentheses.

    For x y plus 2 x plus 3 y plus 6, you can factorise out open parentheses y plus 2 close parentheses.

    Both end up with open parentheses x plus 3 close parentheses open parentheses y plus 2 close parentheses.

  • Define a quadratic expression.

    A quadratic expression is an expression of the form a x squared plus b x plus c where a not equal to 0.

    E.g. x squared plus 6 x minus 2 is a quadratic expression.

  • An expression x squared plus 8 x plus 12 factorises to open parentheses x plus p close parentheses open parentheses x plus q close parentheses.

    Explain how the numbers p and q relate to the numbers 8 and 12.

    If x squared plus 8 x plus 12 factorises to open parentheses x plus p close parentheses open parentheses x plus q close parentheses, then:

    • p plus q equals 8 (the numbers must add to give 8).

    • p q equals 12 (the numbers must multiply to give 12).

  • True or False?

    If x squared minus 54 x plus 288 factorises to open parentheses x plus p close parentheses open parentheses x plus q close parentheses then p and q must both be negative.

    True.

    If x squared minus 54 x plus 288 factorises to open parentheses x plus p close parentheses open parentheses x plus q close parentheses, then p and q multiply to give 288, which is positive. That means that p and q could both be positive or both be negative.

    But since p and q add to give -54 which is negative, then at least one of them is negative.

    The two facts above mean that p and q are both negative.

  • True or False?

    A quadratic expression will always factorise into double brackets.

    False.

    The quadratic expression x squared plus 5 x with no constant term factorises to x open parentheses x plus 5 close parentheses which is not a double bracket expansion.

  • True or False?

    To factorise 3 x squared minus 5 x minus 2, it could first be written in the form 3 x squared plus x minus 6 x minus 2.

    True.

    To factorise harder quadratics like 3 x squared minus 5 x minus 2, you can:

    1. Multiply the first and last numbers together, 3 cross times open parentheses negative 2 close parentheses equals negative 6

    2. Find two numbers that add to the x coefficient, -5, and multiply to -6, i.e. 1 and -6

    3. Split the middle term into 1 x and negative 6 x

    3 x squared minus 5 x minus 2 equals 3 x squared plus x minus 6 x minus 2.

    This then helps to factorise: x open parentheses 3 x plus 1 close parentheses minus 2 open parentheses 3 x plus 1 close parentheses giving open parentheses 3 x plus 1 close parentheses open parentheses x minus 2 close parentheses.

  • True or False?

    open parentheses 5 x plus 2 close parentheses open parentheses 3 minus x close parentheses and open parentheses 2 minus 5 x close parentheses open parentheses x minus 3 close parentheses are both correct factorised expressions of negative 5 x squared plus 13 x plus 6.

    False.

    open parentheses 5 x plus 2 close parentheses open parentheses 3 minus x close parentheses is a correct factorised expression of negative 5 x squared plus 13 x plus 6.

    However, open parentheses 2 minus 5 x close parentheses open parentheses x minus 3 close parentheses expands to negative 5 x squared plus 17 x minus 6.

  • What should the first step be when factorising the expression 2 x squared plus 8 x plus 6?

    When factorising the expression 2 x squared plus 8 x plus 6, check whether each term in 2 x squared plus 8 x plus 6 has a common factor. There is a common factor of 2.

    So the first step when factorising 2 x squared plus 8 x plus 6 should be to factorise out a 2, to get 2 open parentheses x squared plus 4 x plus 3 close parentheses.

    Do not divide by a 2 and get rid of it; you cannot get rid of numbers in expressions by dividing (you can only do that with equations).

  • Define the difference of two squares when talking about factorisation.

    The difference of two squares says that a squared minus b squared factorises into the double brackets open parentheses a plus b close parentheses open parentheses a minus b close parentheses.

  • True or False?

    x squared minus 81 is open parentheses x plus 9 close parentheses open parentheses x minus 9 close parentheses but not open parentheses x minus 9 close parentheses open parentheses x plus 9 close parentheses.

    False.

    x squared minus 81 can be written as open parentheses x plus 9 close parentheses open parentheses x minus 9 close parentheses or as open parentheses x minus 9 close parentheses open parentheses x plus 9 close parentheses, because they both expand to give x squared minus 81.

  • True or False?

    x squared minus 81 is open parentheses x plus 9 close parentheses open parentheses x minus 9 close parentheses but not open parentheses 9 plus x close parentheses open parentheses 9 minus x close parentheses.

    True.

    x squared minus 81 is open parentheses x plus 9 close parentheses open parentheses x minus 9 close parentheses but not open parentheses 9 plus x close parentheses open parentheses 9 minus x close parentheses. The second one expands to give 81 minus x squared, not x squared minus 81.

  • True or False?

    It is impossible to factorise a quadratic expression with no middle term in x into double brackets.

    False.

    The quadratic expression x squared minus 9 has no middle term in xbut factorises into open parentheses x plus 3 close parentheses open parentheses x minus 3 close parentheses using the difference of two squares.

  • Explain how to use the difference of two squares to factorise 4 x squared minus 25.

    To factorise 4 x squared minus 25 using difference of two squares, the 4 x squared can be thought of as open parentheses 2 x close parentheses squared. So 4 x squared minus 25 is open parentheses 2 x close parentheses squared minus 5 squared.

    The difference of two squares can then be used where a equals 2 x and b equals 5, giving open parentheses 2 x plus 5 close parentheses open parentheses 2 x minus 5 close parentheses.

  • Explain how to use the difference of two squares to factorise 5 x squared minus 45.

    To factorise 5 x squared minus 45 using difference of two squares, first factorise out the 5 to get 5 open parentheses x squared minus 9 close parentheses.

    Then use the difference of two squares for the x squared minus 9 part.

    This gives 5 open parentheses x plus 3 close parentheses open parentheses x minus 3 close parentheses.

  • A calculator gives the solutions to 2 x squared plus 7 x plus 3 equals 0 as negative 3 and negative 1 half.

    Explain why this tells you that 2 x squared plus 7 x plus 3 can be factorised into double brackets.

    If the solutions to a quadratic equation are integers or (rational) fractions, then the quadratic factorises.

    The solutions are negative 3 and negative 1 half which are integers or fractions, so it must factorise.

    The factorisation is 2 x squared plus 7 x plus 3 equals open parentheses 2 x plus 1 close parentheses open parentheses x plus 3 close parentheses.

  • True or False?

    The value of b squared minus 4 a c from the quadratic formula for the equation 2 x squared plus 7 x plus 3 equals 0 is equal to 25.

    Therefore the expression 2 x squared plus 7 x plus 3 can be factorised.

    True.

    If the value of b squared minus 4 a c from the quadratic formula is a positive square number, then the quadratic expression factorises.

    As b squared minus 4 a c equals 25 and 25 is a square number, 2 x squared plus 7 x plus 3 must factorise.

    The factorisation is 2 x squared plus 7 x plus 3 equals open parentheses 2 x plus 1 close parentheses open parentheses x plus 3 close parentheses.

  • True or False?

    Quadratic expressions with only two terms can always be factorised.

    False.

    Quadratic expressions with only two terms can not always be factorised.

    E.g. the quadratic expression x squared plus 4 cannot be factorised.

  • True or False?

    The expression x squared minus 6 x can be simplified to x minus 6 by dividing through by x.

    False.

    x squared minus 6 x is an expression, not an equation, so you cannot divide both sides by x (because there are not two sides!).

    Instead, you can factorise out an x to get x open parentheses x minus 6 close parentheses.

    You cannot simplify this any further.

  • True or False?

    If 3 x squared plus 4 x plus 1 factorises to open parentheses 3 x plus 1 close parentheses open parentheses x plus 1 close parentheses then multiplying by 2 means that 6 x squared plus 8 x plus 2 factorises to open parentheses 6 x plus 2 close parentheses open parentheses 2 x plus 2 close parentheses.

    False.

    If 3 x squared plus 4 x plus 1 factorises to open parentheses 3 x plus 1 close parentheses open parentheses x plus 1 close parentheses then multiplying by 2 means that 6 x squared plus 8 x plus 2 factorises to either open parentheses 6 x plus 2 close parentheses open parentheses x plus 1 close parentheses, where the 2 is taken into the first bracket, or open parentheses 3 x plus 1 close parentheses open parentheses 2 x plus 2 close parentheses, where the 2 is taken into the second bracket.

    You cannot put a 2 into both the first and second brackets to get open parentheses 6 x plus 2 close parentheses open parentheses 2 x plus 2 close parentheses. That would be multiplying 3 x squared plus 4 x plus 1 by 4.

    Both correct versions factorise further to 2 open parentheses 3 x plus 1 close parentheses open parentheses x plus 1 close parentheses.

  • True or False?

    A calculator gives the solutions to x squared minus 4 x plus 1 equals 0 as 2 plus square root of 3 and 2 minus square root of 3.

    This means that the expression x squared minus 4 x plus 1 can be written as open parentheses x minus p close parentheses open parentheses x minus q close parentheses where p and q are integers.

    False.

    If the solutions to a quadratic equation are integers or (rational) fractions, then the quadratic factorises.

    The solutions are 2 plus square root of 3 and 2 minus square root of 3 which are neither integers nor fractions, so it does not factorise.

    x squared minus 4 x plus 1cannot be written in the form open parentheses x minus p close parentheses open parentheses x minus q close parentheses where p and q are integers.