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Look Before You Leap

Age 16 to 18 Challenge Level:

If $a+b+c=4$,
and $abc=3$,

what are the values of:
${1\over a}+ {1\over b }+ {1\over c}$ (think of fractions),

${1\over ab}+ {1\over bc }+ {1\over ca}$

and $a^2 +b^2 + c^2$?

Square of side a+b+c

In the diagram the coloured squares have sides of length $a$, $b$ and $c$. Use the areas in the diagram to write down a formula for the expansion of $(a + b + c)^2$ and explain your method.

Using your expansion of $(a + b + c)^2$ to help you, expand $(a + b + c)^3$. Can you explain each term of the expansion using a diagram of a cube where each face has been cut up in a similar way to the square above?