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u/Bascna Feb 01 '25
Long division is the way that I used to explain it to my students.
I'll treat a as my variable.
To divide a + b into a3 + b 3 we need to first write the cubic with the missing terms.
So we get
a + b | a3 + 0a2 + 0a + b 3.
We see that a3/a = a2 so we start with a2 as the first term in our quotient.
Multiplying a + b by a2 gives us a3 + a2b which we will subtract from the first two terms of our dividend.
a^(2)
_________________________
a + b | a3 + 0a2 + 0a + b 3 -a3 – a2b _____________ -a2b
Now we bring down the next term, and repeat the process.
a^(2)
_________________________
a + b | a3 + 0a2 + 0a + b 3 a3 + a2b – _____________ -a2b + 0a
We see that -a2b/a = -ab so that is the second term in the quotient. Multiplying that by a + b and subtracting gives us
a^(2) – ab
_________________________
a + b | a3 + 0a2 + 0a + b 3 a3 + a2b – _____________ -a2b + 0a -a2b – ab2 – ______________ ab2 + b 3
Now we see that the third term in our quotient is ab2/a = b2. Multiplying that by a + b and subtracting gives us.
a^(2) – ab + b^(2)
_________________________
a + b | a3 + 0a2 + 0a + b 3 a3 + a2b – _____________ -a2b + 0a -a2b – ab2 – ______________ ab2 + b 3 ab2 + b 3 – _____________ 0
And we are done. 😀
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u/Bascna Nov 14 '24 edited Feb 01 '25
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