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9.7 Expanding the product of two linear expressions

6 Expand and simplify each expression.
a (y + 5) b (z + 1) c (m + 8) 2
2
2
d (a − 2) e (p − 4) f (n − 9) 2
2
2
7 a Expand and simplify each expression.
i (x + 2)(x − 2) ii (x − 5)(x + 5) iii (x + 7)(x − 7) 1 2 3 4 5
b What do you notice about your answers in part a? 6 7 8 9 10
c Write down the simplified expansion of (x − 10)(x + 10).
d Write down the simplified expansion of (x − y)(x + y). 11 12 13 14 15
8 Here is part of a number grid. 16 17 18 19 20
Look at the red block of four squares, and follow these steps.
c Multiply the number in the bottom left square by the number 21 22 23 24 25
in the top right square: 9 × 5 = 45
d Multiply the number in the top left square by the number 26 27 28 29 30
in the bottom right square: 4 × 10 = 40.
e Subtract the second answer from the first: 45 − 40 = 5. 31 32 33 34 35
a Repeat these three steps with the blue block of four squares. 36 37 38 39 40
b Repeat these three steps with the green block of four squares.
c What do you notice about your answers to a and b?
d Here is a block of four squares from the same number grid. n
Copy the block of four squares and write an expression, in terms of n, in
each of the other squares to represent the missing numbers.
e Repeat the three steps above with the block of four squares in part d.
What do you notice about your answer?

Summary

You should now know that: You should be able to:
+ To multiply powers of the same variable, add the + Use index notation for positive integer powers;
indices. x × x = x a + b apply the index laws for multiplication and
a
b
+ To divide powers of the same variable, subtract the division to simple algebraic expressions.
indices. x ÷ x = x a − b + Construct algebraic expressions.
b
a
+ The letter that is on its own in a formula is called the + Substitute positive and negative numbers into
subject of the formula. expressions and formulae.
+ Depending on the information you are given and + Derive formulae and, in simple cases, change
the variable that you want to find, you may need the subject; use formulae from mathematics and
to rearrange a formula. This is called changing the other subjects.
subject of the formula. + Simplify or transform expressions by taking out
+ When you factorise an expression you take the single-term common factors.
highest common factor and put it outside the + Expand the product of two linear expressions and
brackets. simplify the resulting expression.

+ To add and subtract algebraic fractions, you use the
same method that you use to add normal fractions.
+ When you multiply two expressions in brackets
together, you must multiply each term in the first
brackets by each term in the second brackets.

9 Expressions and formulae 95