Brackets out, brackets in

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Presentation transcript:

Brackets out, brackets in Risp 3

BRACKETS OUT, BRACKETS IN Pick three different, non-zero integers between -5 and 5 that don’t sum to zero. Place all of the permutations of these numbers in the boxes below.

BRACKETS OUT, BRACKETS IN Multiply them all out. Then add all the results together. Now take this sum: can you factorise it? Compare notes with your colleagues once you have tried to do this. Do you notice anything? Does it matter what the starting list of numbers is? Can you make any conjectures? Can you prove these?

BRACKETS OUT, BRACKETS IN 𝑎 𝑏 𝑐 𝑏𝑥2 + (𝑎𝑏 + 𝑐)𝑥 + 𝑎𝑐 𝑎 𝑐 𝑏 𝑐𝑥2 + (𝑎𝑐 + 𝑏)𝑥 + 𝑎𝑏 𝑏 𝑎 𝑐 𝑎𝑥2 + (𝑎𝑏 + 𝑐)𝑥 + 𝑏𝑐 𝑏 𝑐 𝑎 𝑐𝑥2 + (𝑏𝑐 + 𝑎)𝑥 + 𝑎𝑏 𝑐 𝑎 𝑏 𝑎𝑥2 + (𝑎𝑐 + 𝑏)𝑥 + 𝑏𝑐 𝑐 𝑏 𝑎 𝑏𝑥2 + (𝑏𝑐 + 𝑎)𝑥 + 𝑎𝑐 𝟐(𝒂 + 𝒃 + 𝒄) 𝒙𝟐 + 𝟐(𝒂𝒃 + 𝒃𝒄 + 𝒄𝒂 + 𝒂 + 𝒃 + 𝒄)𝒙 + 𝟐(𝒂𝒃 + 𝒃𝒄 + 𝒄𝒂)

BRACKETS OUT, BRACKETS IN 𝟐 𝒂 + 𝒃 + 𝒄 𝒙𝟐 + 𝟐 𝒂𝒃 + 𝒃𝒄 + 𝒄𝒂 + 𝒂 + 𝒃 + 𝒄 𝒙 + 𝟐 𝒂𝒃 + 𝒃𝒄 + 𝒄𝒂 𝟐 𝒂+𝒃+𝒄 𝒙 𝟐 + 𝒂𝒃+𝒃𝒄+𝒄𝒂+𝒂+𝒃+𝒄 𝒙+ 𝒂𝒃+𝒃𝒄+𝒄𝒂 𝟐 𝒂+𝒃+𝒄 𝒙 𝟐 + 𝒂𝒃+𝒃𝒄+𝒄𝒂 𝒙+ 𝒂+𝒃+𝒄 𝒙+ 𝒂𝒃+𝒃𝒄+𝒄𝒂 𝟐 𝒂+𝒃+𝒄 𝒙 𝟐 +𝒙 + 𝒂𝒃+𝒃𝒄+𝒄𝒂 𝒙+𝟏 𝟐 𝒂+𝒃+𝒄 𝒙 𝒙+𝟏 + 𝒂𝒃+𝒃𝒄+𝒄𝒂 𝒙+𝟏 𝟐 𝒂+𝒃+𝒄 𝒙+ 𝒂𝒃+𝒃𝒄+𝒄𝒂 𝒙+𝟏 So 𝒙+𝟏 is always a factor.

Resources

BRACKETS OUT, BRACKETS IN Pick three different, non-zero integers between -5 and 5 that don’t sum to zero. Place all of the permutations of these numbers in the boxes below. SIC_9

BRACKETS OUT, BRACKETS IN Pick three different, non-zero integers between -5 and 5 that don’t sum to zero. Place all of the permutations of these numbers in the boxes below. SIC_9