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GCSEMathsAQAHigherFlashcards2. AlgebraExpanding Brackets

Expanding Brackets (AQA GCSE Maths)

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  • Explain how to expand a single bracket.

    E.g. 2 open parentheses x plus 3 close parentheses.

    Expanding a bracket means multiplying the term outside the bracket by each term inside the bracket.

    So 2 open parentheses x plus 3 close parentheses becomes 2 x plus 6 from 2 cross times x plus 2 cross times 3.

  • True or False?

    negative 2 open parentheses 1 minus x close parentheses expands to negative 2 minus 2 x.

    False.

    negative 2 open parentheses 1 minus x close parentheses can be thought of as open parentheses negative 2 close parentheses cross times 1 plus open parentheses negative 2 close parentheses cross times open parentheses negative x close parentheses.

    Multiplying two negatives gives a positive, so the answer is negative 2 plus 2 x.

  • How can you simplify 3 open parentheses x plus 5 plus x close parentheses before expanding the brackets?

    You can simplify 3 open parentheses x plus 5 plus x close parentheses before expanding by first collecting like terms inside the brackets to give 3 open parentheses 2 x plus 5 close parentheses.

    This can also be written as 3 open parentheses 5 plus 2 x close parentheses.

  • Explain how to simplify an expression with two sets of brackets

    E.g. 2 open parentheses x plus 1 close parentheses plus 3 open parentheses x minus 1 close parentheses.

    To simplify expressions like 2 open parentheses x plus 1 close parentheses plus 3 open parentheses x minus 1 close parentheses you expand the brackets then collect like terms.

    So 2 open parentheses x plus 1 close parentheses plus 3 open parentheses x minus 1 close parentheses expands to give 2 x plus 2 plus 3 x minus 3.
    The terms are then collected to give 5 x minus 1.

  • True or False?

    3 open parentheses x plus 2 close parentheses expands to 3 x plus 6, then the 3's cancel to give x plus 2.

    False.

    You cannot cancel the 3's because 3 open parentheses x plus 2 close parentheses is an expression, not an equation.

    You can only cancel both sides by 3 if you had an equation, like 3 open parentheses x plus 2 close parentheses equals 9.

  • How do you find the highest power of x when expanding expressions?

    E.g. 2 x squared open parentheses x plus x squared close parentheses.

    The highest power of x will come from multiplying the outside term by the inside term with the highest power.

    The highest power of x in 2 x squared open parentheses x plus x squared close parentheses is x to the power of 4.

    This can be found by multiplying x squared outside the bracket by the highest power, x squared, from inside the bracket.

  • When expanding double brackets, e.g. open parentheses x plus 2 close parentheses open parentheses x minus 3 close parentheses, what does the acronym FOIL stand for?

    When expanding double brackets, e.g. open parentheses x plus 2 close parentheses open parentheses x minus 3 close parentheses, the acronym FOIL is used to identify which terms should be multiplied together.

    • F = First (the first term from each bracket, e.g. x cross times x equals x squared)

    • O = Outside (the outer term from each bracket, e.g. x cross times negative 3 equals negative 3 x)

    • I = Inside (the inner term from each bracket, e.g. 2 cross times x equals 2 x)

    • L = Last (the last term from each bracket, e.g. 2 cross times negative 3 equals negative 6)

  • True or False?

    When expanding double brackets such as open parentheses x plus 1 close parentheses open parentheses x plus 2 close parentheses, every term in the left bracket gets multiplied by every term in the right bracket.

    True.

    When expanding double brackets like open parentheses x plus 1 close parentheses open parentheses x plus 2 close parentheses, every term in the left bracket gets multiplied by every term in the right bracket.

    E.g. open parentheses x plus 1 close parentheses open parentheses x plus 2 close parentheses equals x cross times x plus x cross times 2 plus 1 cross times x plus 1 cross times 2.

  • The double brackets open parentheses x plus 1 close parentheses open parentheses x plus 2 close parentheses can be expanded to x squared plus 2 x plus x plus 2.

    What next step is required?

    The next correct step is to simplify the expression by collecting like terms.

    So x squared plus 2 x plus x plus 2 becomes x squared plus 3 x plus 2.

    You cannot simplify it any further.

  • True or False?

    When expanding double brackets using the acronym FOIL, is it ok to rearrange the order, such as LIFO.

    True.

    When expanding double brackets using the acronym FOIL, is it ok to rearrange the order, such as LIFO.

    When you collect terms at the end, you will have an expression that is mathematically the same as using FOIL.

  • True or False?

    open parentheses x plus 5 close parentheses squared equals x squared plus 25.

    False.

    To square a single bracket, you first need to write it as double brackets,
    e.g.open parentheses x plus 5 close parentheses squared equals open parentheses x plus 5 close parentheses open parentheses x plus 5 close parentheses.

    Then expand the double brackets to get x squared plus 5 x plus 5 x plus 25.

    This then simplifies to x squared plus 10 x plus 25.

    The incorrect answer of x squared plus 25 is missing the middle term, 10 x.

  • True or False?

    When expanding open parentheses x plus 1 close parentheses open parentheses x plus 2 close parentheses open parentheses x plus 3 close parentheses, you can choose which pair of brackets to multiply out first (the order does not matter).

    True.

    When expanding open parentheses x plus 1 close parentheses open parentheses x plus 2 close parentheses open parentheses x plus 3 close parentheses, you can choose which pair of brackets to multiply out first (the order does not matter).

  • The expression open parentheses x plus 1 close parentheses open parentheses x plus 2 close parentheses open parentheses x plus 3 close parentheses can be written as open parentheses x plus 1 close parentheses open parentheses x squared plus 3 x plus 2 x plus 6 close parentheses.

    What should be the next step, before expanding any more brackets?

    The expression should be simplified by collecting like terms in the second bracket.

    So open parentheses x plus 1 close parentheses open parentheses x squared plus 3 x plus 2 x plus 6 close parentheses becomes open parentheses x plus 1 close parentheses open parentheses x squared plus 5 x plus 6 close parentheses.

    This reduces the next expansion from 8 calculations to 6 calculations!

  • True or False?

    The x cubed term in the expansion of open parentheses x plus 1 close parentheses open parentheses 2 x plus 1 close parentheses open parentheses 1 plus 3 x close parentheses can be determined without expanding fully.

    True.

    You can see that the x cubed term in the expansion of open parentheses x plus 1 close parentheses open parentheses 2 x plus 1 close parentheses open parentheses 1 plus 3 x close parentheses will be 6 x cubed by:

    1. Writing the brackets in order, open parentheses x plus 1 close parentheses open parentheses 2 x plus 1 close parentheses open parentheses 3 x plus 1 close parentheses

    2. Multiplying the x terms together, x cross times 2 x cross times 3 x

    This gives 6 x cubed, without needing to expand fully.

  • Explain the process for expanding an expression such as open parentheses x minus 2 close parentheses cubed.

    To expand open parentheses x minus 2 close parentheses cubed, you need to:

    1. Write it as a triple bracket expansion, open parentheses x minus 2 close parentheses open parentheses x minus 2 close parentheses open parentheses x minus 2 close parentheses.

    2. Pick two brackets and expand and simplify them.

    3. Expand this result with the remaining bracket.

    4. Simplify the final answer by collecting like terms.

Previous:Algebraic Roots & IndicesNext:Factorising

    1. Topic 1

      Number Operations

    2. Topic 2

      Prime Factors, HCF & LCM

    3. Topic 3

      Powers, Roots & Standard Form

    4. Topic 4

      Fractions

    5. Topic 5

      Percentages

    6. Topic 6

      Simple & Compound Interest, Growth & Decay

    7. Topic 7

      Fractions, Decimals & Percentages

    8. Topic 8

      Rounding, Estimation & Bounds

    9. Topic 9

      Surds

    10. Topic 10

      Using a Calculator

    1. Topic 1

      Introduction to Algebra

    2. Topic 2

      Algebraic Roots & Indices

    3. Topic 3

      Expanding Brackets

    4. Topic 4

      Factorising

    5. Topic 5

      Completing the Square

    6. Topic 6

      Algebraic Fractions

    7. Topic 7

      Rearranging Formulas

    8. Topic 8

      Algebraic Proof

    9. Topic 9

      Linear Equations

    10. Topic 10

      Quadratic Equations

    11. Topic 11

      Simultaneous Equations

    12. Topic 12

      Iteration

    13. Topic 13

      Forming & Solving Equations

    14. Topic 14

      Functions

    15. Topic 15

      Coordinate Geometry

    16. Topic 16

      Linear Graphs

    17. Topic 17

      Graphs of Functions

    18. Topic 18

      Equation of a Circle

    19. Topic 19

      Estimating Gradients & Areas under Graphs

    20. Topic 20

      Real-Life Graphs

    21. Topic 21

      Solving Inequalities

    22. Topic 22

      Graphing Inequalities

    23. Topic 23

      Transformations of Graphs

    24. Topic 24

      Sequences

    1. Topic 1

      Ratios

    2. Topic 2

      Problem Solving with Ratios

    3. Topic 3

      Direct & Inverse Proportion

    4. Topic 4

      Standard & Compound Units

    5. Topic 5

      Exchange Rates & Best Buys

    1. Topic 1

      Symmetry & Shapes

    2. Topic 2

      Angles in Polygons & Parallel Lines

    3. Topic 3

      Bearings, Scale Drawing, Constructions & Loci

    4. Topic 4

      Circle Theorems

    5. Topic 5

      Area & Perimeter

    6. Topic 6

      Circles, Arcs & Sectors

    7. Topic 7

      Volume & Surface Area

    8. Topic 8

      Congruence, Similarity & Geometrical Proof

    9. Topic 9

      Area & Volume of Similar Shapes

    10. Topic 10

      Pythagoras & Trigonometry

    11. Topic 11

      Sine, Cosine Rule & Area of Triangles

    12. Topic 12

      3D Pythagoras & Trigonometry

    13. Topic 13

      Vectors

    14. Topic 14

      Transformations

    1. Topic 1

      Introduction to Probability

    2. Topic 2

      Simple Probability Diagrams

    3. Topic 3

      Tree Diagrams

    4. Topic 4

      Combined & Conditional Probability

    1. Topic 1

      Averages, Ranges & Data

    2. Topic 2

      Statistical Diagrams

    3. Topic 3

      Histograms

    4. Topic 4

      Cumulative Frequency & Box Plots

    5. Topic 5

      Scatter Graphs & Correlation