Question

How do you factor the trinomial `21x^2-40xy+11y^2`?

Answer 1

`color(green)((3x-y)(7x-11y))`

Explanation:

The factors for 21 are {3, 7}, {1, 21}
The factors for 11 are {1, 11}

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A quick mental check shows that {1, 21} for 21 will cause problems for the rest so it is rejected. Consequently most of the work is already done for you!

For `21x^2` It has to be either `{-3x xx -7x} "or" {+3x xx +7x}`

I am going to choose:

`(3x +?)(7x+?)`

Now lets look at the `11y^2`

This can only be the positive or negative versions of `{1yxx11y}`
`color(brown)("Lets play!")`
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`color(blue)(Try color(white)(.) 1)`

`(3x-11y)(7x-y) = 21x^2 -3xy -77xy +11y^2`
`color(red)("This fails for the target of "-40xy)`

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`color(blue)(Try color(white)(.) 2)`

`color(green)((3x-y)(7x-11y)) = 21x^2 -33xy-7xy+11y^2`

`color(blue)(=21x^2-40xy+11y^2) color(white)(...) color(red)( "It works!")`